A RIEMANN-HILBERT CORRESPONDENCE

FOR UNIT F -CRYSTALS. I

Matthew Emerton and Mark Kisin

Department of Mathematics, University of Michigan, Ann Arbor (M.E.) Department of Mathematics, University of Sydney, Sydney, and SFB 478, Westfälische Wilhelms Universität, Münster (M.K.)

Abstract. Let X be a smooth scheme over a field k containing 𝔽_{q}, where q is a power of some prime p. We study the relationship between constructible 𝔽_{q} sheaves on the étale site of X, and a certain class of quasi-coherent O_{X}-modules equipped with a “unit” Frobenius structure. We show that the two corresponding derived categories are anti-equivalent as triangulated categories, and that this anti-equivalence is compatible with direct and inverse images, tensor products, and certain other operations.

Contents

0. Introduction 1

1. O_{F^{r}}-Modules 5

2. Pull-backs of O_{F^{r}}-Modules 16

3. Push-forwards of O_{F^{r}}-Modules 24

4. Relations Between f_{+} and f^{!} 34

5. Unit O_{F^{r}}-modules 51

6. Locally Finitely Generated Unit O_{F^{r}}-Modules 61

7. O_{F^{r}}-Modules on the Étale Site 69

8. 𝔽_{q}-Sheaves on the Étale Site 73

9. The Functor Sol_{ét} 74

10. The Functor M_{ét} 86

11. Riemann-Hilbert Correspondence 91

12. L-functions 100 Appendix. Duality and the Cartier Operator 103 References 111

0. introduction

Let X be a smooth complex analytic space. One knows that there is an equivalence of categories between the category of local systems of ℂ-vector spaces on X and the category of coherent O_{X}-modules equipped with an integrable connection ∇. A serious defect of this theory is that the category of local systems, and similarly that of modules with connection, is not stable under taking direct images. For example, if f : Y ⟶ X is a proper map of smooth complex analytic spaces, then the higher direct images R^{i}f_{∗}ℂ are guaranteed to be a local system only over the points where f is smooth. Similarly, the connection on the relative De Rham cohomology may be singular at the points where f is not smooth. This defect is remedied by the theory of D-modules, which shows that there is a relationship between the category of constructible sheaves, and the category of regular holonomic D-modules. More precisely, the two corresponding derived categories are equivalent as triangulated categories, and this equivalence respects the “six operations” f^{!}, f_{!}, f^{∗}, f_{∗}, \underline{RHom}^{•},

and ⊗^{𝕃} . This result is known as the “Riemann-Hilbert correspondence”.

The purpose of this paper is to study a certain characteristic p analogue of the above situation. Our starting point is a theorem of Nick Katz [Ka, Prop. 4.1.1].

Theorem. (Katz) Let k be a perfect field of characteristic p containing 𝔽_{q}, where q = p^{r}. If X is a smooth scheme over W_{n}(k), the ring of Witt vectors of length n, and if F_{X} is a lift to X of the Frobenius on W_{n}(k), then there is an equivalence of categories between the category of locally free étale sheaves of W_{n}(𝔽_{q})-modules L, and the category of coherent, locally free O_{X}-modules E equipped with an O_{X} linear isomorphism (F_{X}^{r})^{∗}E ⟶^{∼} E. This equivalence is realized by associating E = L ⊗_{W_{n}(k)} O_{X} to L.

In the context of this paper, Katz’s theorem should be regarded as the analogue of the relation between local systems and modules with connection. In the present paper we study only the situation where X is actually a smooth k-scheme. The more general situation is the subject of the paper [EK] (see below).

To explain our results, we introduce the notion of a locally finitely generated unit O_{F^{r},X}-module. If r is a fixed positive integer, than a unit O_{F^{r},X}-module on X is a quasi-coherent O_{X}-module M equipped with an isomorphism (F_{X}^{r})^{∗}M ⟶^{∼} M. This isomorphism endows M with the structure of a sheaf of modules over the sheaf of rings O_{F^{r},X} = O_{X}[F^{r}], where O_{X}[F^{r}] denotes the twisted polynomial ring given by the relation F^{r}a = a^{q}F^{r}, for any section a of O_{X}. The unit O_{F^{r},X}-module M is said to be locally finitely generated if, locally on X, it is finitely generated as an O_{F^{r},X}-module.

The main result of this paper is the Riemann-Hilbert correspondence of its title, which generalizes Katz’s theorem to an (anti-) equivalence of two triangulated categories: the derived category D_{c}^{b}(X_{ét}) of bounded complexes of étale sheaves of 𝔽_{q}-modules whose cohomology sheaves are constructible, and the derived category D_{lfgu}^{b}(O_{F^{r},X}) of bounded complexes of O_{F^{r},X}-modules (on the Zariski site of X) whose cohomology sheaves are locally finitely generated unit O_{F^{r},X}-modules.

If we let π_{X} : X_{ét} → X denote the natural morphism from the étale site to the Zariski site of X, then this anti-equivalence is given by associating

M^{•} = M(F^{•}) = Rπ_{X∗}\underline{RHom}_{𝔽_{q}}^{•}(F^{•}, O_{X})[d_{X}]

to a complex F^{•} in D_{c}^{b}(X_{ét}), and associating

F^{•} = Sol(M^{•}) = \underline{RHom}_{O_{F^{r} ,X}}^{•} (π_{X}^{∗}M^{•}, O_{Xét})[d_{X}]

to a complex M^{•} in D_{lfgu}^{b}(O_{F^{r},X}). (Here d_{X} denotes the (locally constant) dimension of X.) A comment on the choice of notation: M is intended to suggest a Dieudonné module functor, while Sol is borrowed from the theory of D-modules.

The correspondence we construct is compatible with three of Grothendieck’s six

operations, namely the operations f_{!}, f^{−1}, and ⊗^{𝕃} on D_{c}^{b}(X_{ét}). (Of course, for this claim of compatibility to be well-formed, one must have an a priori description of the corresponding triple of operators on D_{lfgu}^{b}(O_{F^{r},X}). A significant part of our work is devoted to constructing such operations for O_{F^{r},X}-modules and establishing their main properties.) This may at first seem disappointing, but in fact these are the only three which are defined in our situation. Indeed for complexes of ptorsion étale sheaves the push-forward f_{∗} does not preserve the property of having constructible cohomology sheaves. Also one cannot define f^{!}, since there is no duality (for example because the purity theorem fails horribly). This lack of a duality is also the reason we construct an anti-equivalence rather than an equivalence of categories.

As a compensation, we show that our correspondence relates two other operations, induction and restriction, which have no analogue over ℂ. Namely, if q′ = p^{r′}, and 𝔽_{q} ⊂ 𝔽_{q′} ⊂ k, then the functor – ⊗_{𝔽_{q}} 𝔽_{q′} converts sheaves of 𝔽_{q}-vector spaces into sheaves of 𝔽_{q′}-vector spaces. Conversely the functor “pass to the underlying 𝔽_{q}-structure” converts sheaves of 𝔽_{q′}-vector spaces into sheaves of 𝔽_{q}-vector spaces. We call these operations respectively induction and restriction. There are corresponding operations for O_{F^{r},X}-modules, namely – ⊗_{O_{F^{r′} ,X}} O_{F^{r},X} and “pass to the underlying O_{F^{r′} ,X}-module structure”, which we also call “induction” and “restriction”. We show that our anti-equivalence interchanges the operations of induction and restriction.

Also, although in the statement of Katz’s theorem k denotes a perfect field, in the body of the text we allow k to be an arbitrary field of characteristic p, and a technique of inseparable descent is included in our theory, in a manner compatible with our Riemann-Hilbert correspondence.

The final “operation” that our Riemann-Hilbert correspondence respects is the formation of L-functions (when k is a finite field, so that these are defined).

As mentioned above, the sequel [EK] to this paper will deal with the situation of schemes X which are smooth over W_{n}(k). The results in this more general situation use all the theory developed in the present paper, but include significant extra features. We no longer work with the simple sheaf of rings O_{F^{r},X} but with a more intricate sheaf of rings D_{F^{r},X}, obtained by adjoining to O_{X} the differential operators of all orders on X together with the r^{th} power of a “local lift of Frobenius.” A pleasing point is that the sheaf D_{F^{r},X} that one obtains is independent of the choice of local lifts, as any two lifts are congruent modulo p, and so either lift may be expressed as a polynomial combination of the other lift and appropriate differential operators. Thus (unlike in Katz’s result stated above) we are not restricted to the consideration of smooth W (k)-schemes which admit a global lift of Frobenius. Another important technical result is that locally the sheaf O_{X} has a finite length resolution by free D_{F^{r},X}-modules, whereas this is false if one works just with differential operators, and does not adjoin the Frobenius lifts. Thus the differential operators and lifts of Frobenius complement one another.

Our original motivation for writing this paper and its sequel was to try to generalize the results (due to Fontaine, Messing, Kato, Faltings, Tsuji ...) relating p-adic étale cohomology and crystalline cohomology of schemes over a finite extension of ℚ_{p} to a Riemann-Hilbert type correspondence. That such a theory should exist is strongly suggested by the work of Faltings [Fa], where he treats the correspondence between étale and crystalline cohomology allowing coefficients. We intended the present paper and its sequel to be a “warm-up” for this more general project, but as the reader will see, even the comparatively simple unit case has turned out to be quite technical.

After we had begun this work, we were pleasantly surprised to find that a correspondence of the type we prove here had been recently suggested by Lyubeznik [Lyu, p. 69] (though not at the level of derived categories). As the reader will see, we are indebted to him for providing several important techniques for working with unit O_{F^{r},X}-modules, and especially for the notion of a “generator” of such a module.

We hope that our results will have applications in various directions. One application already found has been to a conjecture of Katz (stated in [Ka2]) relating L-functions of p-adic lisse sheaves on varieties over finite fields of characteristic p to their cohomology with proper supports. Using the techniques of this paper, this conjecture has now been proved [EK2]. There is also the possibility that the formalism developed here will have applications to the study of local cohomology on varieties in characteristic p. Indeed, in the case that X is a smooth affine variety over k, our notion of locally finitely generated unit O_{F^{r},X}-module coincides with the notion of F -finite module, which was introduced by Lyubeznik [Lyu] expressly for this purpose.

We now give a more detailed outline of our paper. It naturally falls into three portions. The first, consisting of sections 1, 2, 3 and 4, introduces the notions of O_{F^{r}}-modules, defines the functors f^{!} and f_{+}, and develops their basic properties. The approach is modelled closely on the theory of algebraic D-modules, and as in that theory, there is a close relationship between the properties of the functors f^{!} and f_{+} and the Grothendieck-Serre duality theory of quasi-coherent sheaves.

The second portion, consisting of sections 5 and 6, introduces the notions of unit and locally finitely generated unit O_{F^{r}}-modules. The results of the first four sections are specialized to and further developed in this particular context. (It is an interesting question as to whether those results will have applications outside the context of unit O_{F^{r}}-modules which is the focus of this paper.) One result which is indispensable in the study of locally finitely generated unit O_{F^{r}}-modules is the fact that any such module M on a k-scheme X admits a presentation of the form

0 → O_{F^{r},X} ⊗_{O_{X}} M → O_{F^{r},X} ⊗_{O_{X}} M → M → 0,

where M is a coherent O_{X}-module. The idea of using such presentations to study locally finitely generated unit O_{F^{r},X}-modules is an extension of the main technique of [Lyu].

Sections 7 and 8 provide a bridge between the first two portions and the third. Section 7 observes that the entire theory of sections 1 through 6 could have been developed by working on the étale site X_{ét} of a smooth k-scheme X rather than on the Zariski site. Indeed, both theories could have been developed in parallel, but we chose not to do so for expository reasons; it seemed easier to fix ideas by working on one site or the other. A key observation is that, if one restricts oneself to quasicoherent O_{F^{r},X}-modules, then étale descent provides an equivalence between the Zariski and étale theories. Section 8 recalls (very briefly) the theory of constructible 𝔽_{q}-sheaves on X_{ét}, as developed in [SGA 4] and [De].

The final portion of the paper consists of section 9, 10, 11 and 12. In these four sections we establish our Riemann-Hilbert correspondence. The first three sections are devoted to defining the functors M and Sol, proving that they yield the desired equivalence of categories, and showing that they respect the appropriate Grothendieck operations. Section 12 then provides a brief discussion of L-functions in the case that k is a finite field, and shows that the Riemann-Hilbert correspondence is compatible with their formation. It also includes the statement of the main result of [EK2].

Just as in the case of the Riemann-Hilbert correspondence for D-modules, the proof of the Riemann-Hilbert correspondence of this paper uses excision arguments to reduce to an explicit simple fact. In the case of D-modules, it is the fact that any function on a smooth complex variety with vanishing partial derivatives is locally constant. In our situation, it is the fact that any section of the structure sheaf on a smooth k-scheme which is F^{r}-invariant is locally constant and 𝔽_{q}-valued. So just as the Riemann-Hilbert correspondence for D-modules can be regarded as a generalization of De Rham theory, our Riemann-Hilbert correspondence can be regarded as a generalization of Artin-Schreier theory.

One consequence of our Riemann-Hilbert correspondence worth mentioning here is that it induces a non-standard t-structure on the category D_{c}^{b}(X_{ét}) for any smooth k-scheme X. At the present moment we have no intrinsic description of this tstructure.

In setting up a cohomological formalism of the type that we have, there are invariably a large number of compatibilities that can and should be checked between the various functors and natural transformations that one defines. Our approach to these has been as follows: those compatibilities which are essential for establishing the main results are stated and either proved or left to the reader in the case that the verifications are standard (if somewhat lengthy). In doing this our aim has been to make the arguments clear and complete, while keeping the size of the paper bounded. The construction and verification of the many other compatibilities that we have not discussed are left to the imagination of the reader.

On a note related to that of the preceding paragraph, it is important to draw attention to two foundational issues on which this paper depends. The first is that we need some consistent choice of sign conventions for the various homological constructions that we use. The second is that much of our theory depends (not just on the statements of, but) on the construction of Grothendieck-Serre duality, and the standard reference [Ha] for this construction is incomplete in its details. Thankfully, both issues are resolved by a recent preprint of Brian Conrad [Con]; in this very detailed paper, Conrad lays down a consistent set of sign conventions, and also provides an extremely close reading of [Ha], filling in many gaps and correcting various misstatements. Throughout this paper we will follow [Con] with regard to both its sign conventions (although these will not play any overt role) and its analysis of [Ha].

In the appendix we have provided a summary of the results of Grothendieck-Serre duality that we use in the paper, and have developed some consequences of the general theory which are of particular importance for our purposes. In particular, we give a definition of the Cartier operator in a broader context than that in which it is usually considered.

1. O_{F^{r}}-modules

(1.1) We fix a prime p > 0, a positive integer r, and set q = p^{r}. Let k be a field containing 𝔽_{q}, and let X be a scheme over k (throughout this paper, the phrases “scheme over k” and “k-scheme” will mean “finite type separated kscheme”, whether or not these hypotheses are explicitly mentioned). We are going to define a sheaf O_{F^{r},X} on X.

Definition 1.1.1. If A is any k-algebra, we define A [F^{r}] to be the twisted polynomial ring over A whose multiplication is defined by F^{r}a = a^{q}F^{r}. If X is a scheme over k, we can sheafify this definition, to obtain a sheaf of rings O_{X}[F^{r}] on X, whose sections over any open subset U are O_{X}[F^{r}](U) = O_{U}(U)[F^{r}]. We often denote O_{X}[F^{r}] by O_{F^{r},X}.

(1.2) Recall that if X is a k-scheme then the (absolute) Frobenius morphism F_{X} : X → X is defined to be the identity on the underlying space of X; the map

F_{X}^{#} : O_{X} → F_{X∗}O_{X} = O_{X}

is defined by a ↦ a^{p}.

Definition 1.2.1. For any non-negative integer n we let O_{X}^{(n)} denote the sheaf O_{X} regarded as an (O_{X}, O_{X})-bimodule in the following way: for the left O_{X}-module structure, O_{X} is regarded as a module over itself in the usual way, while for the right O_{X}-module structure, O_{X} is regarded as a module over itself via the morphism (F_{X}^{#})^{n}. Thus any section a of O_{X} acts on O_{X}^{(n)} on the left as multiplication by a, and on the right as multiplication by a^{p^{n}}.

If M is any O_{X}-module, then we have a natural isomorphism

(1.2.2)  (F_{X}^{n})^{∗}M ⟶^{∼} O_{X}^{(n)} ⊗_{O_{X}} M.

(1.3) Since O_{X} is a sheaf of subrings of O_{F^{r},X} = O_{X}[F^{r}], we may naturally regard O_{F^{r},X} as an (O_{X}, O_{X})-bimodule.

Lemma 1.3.1. There is an isomorphism of (O_{X}, O_{X})-bimodules

⨁_{n=0}^{∞} O_{X}^{(rn)} ⟶^{∼} O_{F^{r},X},

which is given in degree n by the formula a ↦ aF^{rn}. In particular, when X is smooth over k, O_{F^{r},X} is locally free as both a left and a right O_{X}-module.

Proof. The claimed isomorphism is immediate from the definition of the ring structure of O_{F^{r},X} and the bimodule structure of O_{X}^{(n)}. That O_{F^{r},X} is free as a left O_{X} is now clear, and the analogous claim for the right O_{X}-module structure follows from the fact that because X is smooth, each O_{X}^{(n)} is locally free as a right O_{X}-module (see (A.2)). □

Corollary 1.3.2. If M is a sheaf of O_{X}-modules, there is a natural isomorphism of left O_{X}-modules

⨁_{n=0}^{∞}(F_{X}^{rn})^{∗}M ⟶^{∼} O_{F^{r},X} ⊗_{O_{X}} M.

Proof. This follows from the lemma and equation (1.2.2). □

(1.4) The natural isomorphism O_{X} ⟶^{∼} Hom_{O_{X}} (O_{X}, O_{X}) (given by a section a of O_{X} acting as multiplication by a) extends to a map

O_{F^{r},X} → Hom_{𝔽_{q}}(O_{X}, O_{X})

defined by taking F^{r} to the homomorphism f ↦ f^{q}. In future we will always regard O_{X} with the left O_{F^{r},X}-module structure given by this map.

This map need not be an injection (despite a remark suggesting the contrary on page 102 of [Lyu], where A [F] is denoted by A{f}); for example if r = 1, and X = Spec k, then the global section F − 1 is in the kernel; if X = Spec k [x, y]/(x^{2}, xy), then the global section x − xF is in the kernel. The following lemma clarifies the situation:

Lemma 1.4.1. If X is a reduced k-scheme having each irreducible component of positive dimension, then the map

O_{F^{r},X} → Hom_{𝔽_{q}}(O_{X}, O_{X})

is injective.

Proof. We may assume that r = 1, and that X = Spec A is affine, so that A is a reduced finite type k-algebra. Let x be a closed point of X. Since each irreducible component of X is of positive dimension, there exists a non-zero divisor a ∈ A contained in the maximal ideal 𝔪 of x. Indeed, otherwise 𝔪 is in the union of the minimal primes of A (since A is reduced, all the associated primes of A are minimal), hence is contained in one of them, which contradicts the assumption that the components of X have positive dimension.

Now suppose that ∑_{i=0}^{n} a_{i}F^{n} is the zero endomorphism of A. Then for any positive integer m,

0 = ∑_{i=0}^{n} a_{i}F^{n}(a^{m}) = ∑_{i=0}^{n} a_{i}a^{mp^{n}}.

But from this equation, cancelling a power of a^{m}, we conclude that a_{0} is divisible by arbitrarily large powers of a. Thus a_{0} vanishes in a neighborhood of x. Continuing in the same manner, we find that each a_{i} vanishes in a neighborhood of x. Since x was an arbitrary closed point of X, and X is of finite type over k, we see that each a_{i} vanishes on X. □

(1.5) Suppose that M is any sheaf of left O_{F^{r},X}-modules. Multiplication on the left by sections of O_{F^{r},X}-yields an O_{F^{r},X}-linear, and in particular O_{X}-linear, morphism

(1.5.1)  O_{F^{r},X} ⊗_{O_{X}} M → M.

By corollary 1.3.2 there is a natural isomorphism

O_{F^{r},X} ⊗_{O_{X}} M ⟶^{∼} ⨁_{n=0}^{∞}(F_{X}^{rn})^{∗}M,

and so the morphism (1.5.1) determines, and is determined by, a series of O_{X}-linear morphisms

ϕ_{n,M} : (F_{X}^{rn})^{∗}M = O_{X}^{(rn)} ⊗_{O_{X}} M → M

(n a non-negative integer). Explicitly, ϕ_{n,M} is given by the formula

ϕ_{n,M} : a ⊗ m ↦ aF^{rn}(m),

where a is a section of O_{X}^{(rn)} and m a section of M.

Lemma 1.5.2. Suppose that M is a sheaf of left O_{F^{r},X}-modules, and let

ϕ_{n,M} : (F_{X}^{rn})^{∗}M → M

be the sequence of O_{X}-linear morphisms constructed above. Then ϕ_{0,M} = id_{M}, while for each n > 0, ϕ_{n,M} is equal to the composition

(F_{X}^{rn})^{∗}M ⟶^{(F_{X}^{rn−r})^{∗}ϕ_{1,M}} (F_{X}^{rn−r})^{∗}M ⟶^{(F_{X}^{rn−2r})^{∗}ϕ_{1,M}} ⋯ ⟶^{F_{X}^{r}^{∗}ϕ_{1,M}} F_{X}^{r}^{∗}M ⟶^{ϕ_{1,M}} M.

Conversely, if M is a sheaf of O_{X}-modules equipped with a morphism ϕ_{1} : F_{X}^{r}^{∗}M → M, and one defines ϕ_{0} = id_{M} and, for each n > 0, defines ϕ_{n} to be the composition

(F_{X}^{rn})^{∗}M ⟶^{(F_{X}^{rn−r})^{∗}ϕ_{1}} (F_{X}^{rn−r})^{∗}M ⟶^{(F_{X}^{rn−2r})^{∗}ϕ_{1}} ⋯ ⟶^{F_{X}^{r}^{∗}ϕ_{1}} F_{X}^{r}^{∗}M ⟶^{ϕ_{1}} M,

then there is a unique left O_{F^{r},X}-module M having M as underlying O_{X}-module and for which ϕ_{n,M} = ϕ_{n} for each non-negative integer n.

Proof. The fact that ϕ_{0,M} = id_{X} is equivalent to the fact that the section 1 of O_{F^{r},X} acts identically on M, and the composition formula for ϕ_{n,M} (n > 0) is equivalent to the associative law for the action of O_{F^{r},X} on M. Conversely, suppose given an O_{X}-linear morphism ϕ : F_{X}^{r}^{∗}M → M. For any section m of M, define F^{r}(m) = ϕ(1⊗m). This defines the required left O_{F^{r},X}-module structure on M. □

Definition 1.5.3. If M is a left O_{F^{r},X}-module, we write

ϕ_{M} = ϕ_{1,M} : F_{X}^{r}^{∗}M → M,

defined by a ⊗ m ↦ aF^{r}(m), and refer to this as the structural morphism of M.

The previous lemma shows that M is uniquely determined by its structural morphism. Example 1.5.4 The structural morphism of the O_{F^{r},X}-module O_{X} is the natural

isomorphism F_{X}^{r}^{∗}O_{X} ⟶^{∼} O_{X}.

Lemma 1.5.5. Suppose that ψ : M → N is a morphism of left O_{F^{r},X}-modules. Then we have ψ ∘ ϕ_{M} = ϕ_{N} ∘ F_{X}^{r∗}ψ. Conversely, any morphism of O_{X}-modules ψ for which this formula holds is a morphism of left O_{F^{r},X}-modules.

Proof. This is immediate. □

(1.6) Our main object of study for most of this paper will be sheaves of left O_{F^{r},X}-modules on X which are quasi-coherent as O_{X}-modules. We will refer to such a sheaf as a quasi-coherent left O_{F^{r},X}-module. We let μ(X) denote the abelian category of quasi-coherent left O_{F^{r},X}-modules.

We will consider various derived categories of left O_{F^{r},X}-modules. We denote by D^{•}(O_{F^{r},X}) the derived category of complexes of left O_{F^{r},X}-modules, and by D^{•}(μ(X)) the derived category of complexes of quasi-coherent left O_{F^{r},X}-modules. Here • denotes any of the usual boundedness conditions +, −, b, or ∅.

Definition 1.6.1. We let D_{qc}^{b}(O_{F^{r},X}) denote the full sub-triangulated category of D^{b}(O_{F^{r},X}) consisting of complexes whose cohomology sheaves are quasi-coherent left O_{F^{r},X}-modules.

We have inclusions of full triangulated subcategories:

D^{b}(μ(X)) → D_{qc}^{b}(O_{F^{r},X}) → D^{b}(O_{F^{r},X}).

Theorem 1.6.2. The inclusion D^{b}(μ(X)) → D_{qc}^{b}(O_{F^{r},X}) is an equivalence of triangulated categories.

Proof. This follows from Bernstein’s theorem [Bor, VI 2.10]. □

(1.7) The following results provide us with resolutions which will be useful in what is to come.

Lemma 1.7.1. If X is a smooth k-scheme, then any quasi-coherent left O_{F^{r},X}- module is the quotient of a locally free left O_{F^{r},X}-module. Thus any object of D_{qc}^{b}(O_{F^{r},X}) may be represented by a complex bounded above of locally free left O_{F^{r},X}-modules.

Proof. Let M be an object of μ(X). Since X is smooth, there is an O_{X}-linear surjection E → M for some locally free O_{X}-module E ([Ha2, III Ex. 6.8]). Thus we get a surjection O_{F^{r},X} ⊗_{O_{X}} E → M, and the induced module O_{F^{r},X} ⊗_{O_{X}} E is a locally free O_{F^{r},X}-module. The lemma now follows via standard homological techniques from this remark together with Bernstein’s theorem, since a locally free left O_{F^{r},X}-module is a quasi-coherent O_{F^{r},X}-module. □

Corollary 1.7.2. If X is a smooth k-scheme, then any object of D_{qc}^{b}(O_{F^{r},X}) may be represented by a bounded complex of quasi-coherent left O_{F^{r},X}-modules which are flat as O_{X}-modules.

Proof. Let M^{•} be an object of D_{qc}(O_{F^{r},X}). Lemma 1.7.1 shows that we may represent M^{•} via a complex P^{•} bounded above of locally free left O_{F^{r},X}-modules. Now since X is smooth, any quasi-coherent O_{X}-module has finite Tor-dimension. This is true in particular of the finitely many non-vanishing cohomology groups of P^{•}. A spectral sequence argument shows that P^{•} itself is of finite Tor-dimension when regarded as a complex of O_{X}-modules. Standard arguments now show that some finite truncation of P^{•} is a complex of flat O_{X}-modules. (See [Ha, II 4.2].) □

(1.8) We wish to explain a certain convention regarding our notation for shifts. Suppose that X is a k-scheme; then X is the union of finitely many connected components:

X = ∐_{i=1}^{n} X_{i}

with each X_{i} a connected open subscheme of X. Let ψ : X → ℤ be a continuous integer valued function on X. Then ψ is constant on each X_{i}; let ψ_{i} denote the value assumed by ψ on X_{i}.

Then

D_{qc}^{b}(O_{F^{r},X}) = ∏_{i=1}^{n} D_{qc}^{b}(O_{F^{r},X_{i}}).

If M^{•} is an object of D_{qc}^{b}(O_{F^{r},X}), write M^{•} = (M_{1}^{•}, …, M_{n}^{•}), where each M_{i}^{•} is an object of D_{qc}^{b}(O_{F^{r},X_{i}}). Then we define

M^{•}[ψ] = (M^{•}[ψ_{1}], …, M^{•}[ψ_{n}]).

Similarly we will say that H^{i}(M^{•}) = 0 for i > ψ if H^{i}(M_{n}^{•}) = 0 for i > ψ_{n}. One such ψ which is defined for any k-scheme X is the morphism

d_{X} : x ↦ dimension of component of X containing x.

More generally, if f : Y → X is a morphism between k-schemes, then we may consider the functions d_{Y/X} = d_{Y} − d_{X} ∘ f and d_{X/Y} = −d_{Y/X}.

(1.9) Suppose that M and N are left O_{F^{r},X}-modules, and so in particular O_{X}- modules. Then the tensor product M ⊗_{O_{X}} N is also an O_{X}-module. We extend this O_{X}-module structure to a left O_{F^{r},X}-module structure on M⊗_{O_{X}} N by letting F act diagonally. Thus we obtain a bifunctor

– ⊗_{O_{X}} – : O_{F^{r},X}-Mod × O_{F^{r},X}-Mod → O_{F^{r},X}-Mod.

As the tensor product of quasi-coherent O_{X}-modules is quasi-coherent, this bifunctor restricts to a bifunctor

– ⊗_{O_{X}} – : μ(X) × μ(X) → μ(X).

The following results describe the structural morphism of the tensor product M ⊗_{O_{X}} N , for two left O_{F^{r},X}-modules M and N.

Lemma 1.9.1. If M and N are sheaves of O_{X}-modules on a k-scheme X then there is a natural isomorphism

F_{X}^{r}^{∗}M ⊗_{O_{X}} F_{X}^{r}^{∗}N ⟶^{∼} F_{X}^{r}^{∗}(M ⊗_{O_{X}} N).

Proof. Define a morphism on the level of sections by:

(a ⊗ m) ⊗ (b ⊗ n) ↦ ab ⊗ (m ⊗ n),

for a section a ⊗ m of F_{X}^{r}^{∗}M = O_{X}^{(r)} ⊗_{O_{X}} M and a section b ⊗ n of F_{X}^{r}^{∗}N = O_{X}^{(r)} ⊗_{O_{X}} N. Then it is immediate that this provides the required isomorphism. (This is of course just a special case of a general property of ⊗.) □

Lemma 1.9.2. If M and N are left O_{F^{r},X}-modules then the following diagram, in which the top arrow is the isomorphism of the lemma 1.9.1, commutes:

\xymatrix{F_{X}^{r}^{∗}M ⊗_{O_{X}} F_{X}^{r}^{∗}N \ar[r]^{∼} \ar[dr]^{ϕ_{M}⊗ϕ_{N}} & F_{X}^{r}^{∗}(M ⊗_{O_{X}} N) \ar[d]^{ϕ_{M⊗_{O_{X}} N}} \\  & M ⊗_{O_{X}} N .}

Proof. This is immediate. □

Lemma 1.9.3. If M is a left O_{X}-module, then the natural isomorphism of O_{X}- modules O_{X} ⊗_{O_{X}} M = M, defined by a ⊗ m ↦ am for any sections a and m of O_{X} and M respectively, is an isomorphism of left O_{F^{r},X}-modules.

Proof. This follows immediately from the equations

F^{r}am = a^{q}F m = (F^{r}a)(F^{r}m).

□

(1.9.4) Corollary 1.7.2 allows us to define the left derived functor of – ⊗_{O_{X}} –, which is a bifunctor

– ⊗^{𝕃}_{O_{X}} – : D_{qc}^{b}(O_{F^{r},X}) × D_{qc}^{b}(O_{F^{r},X}) → D_{qc}^{b}(O_{F^{r},X}),

exact in each variable.

We later need the following lemma:

Lemma 1.9.5. If P and Q are both locally free left O_{F^{r},X}-modules, then the tensor product P ⊗_{O_{X}} Q is a locally free left O_{F^{r},X}-module.

Since tensor commutes with direct sum, it suffices to verify this when P = Q = O_{F^{r},X}. In this case we see immediately that O_{F^{r},X} ⊗_{O_{X}} O_{F^{r},X} is freely generated as a left O_{F^{r},X}-module by the countably many sections

1 ⊗ 1, 1 ⊗ F^{r}, F^{r} ⊗ 1, ⋯, 1 ⊗ F^{rn}, F^{rn} ⊗ 1, · · · .

□

Definition 1.10. If M is an O_{X}-module, then O_{F^{r},X} ⊗_{O_{X}} M is a left O_{F^{r},X}- module, which we refer to as the left O_{F^{r},X}-module induced by M.

More generally, if M is a left O_{F^{r},X}-module isomorphic to a left O_{F^{r},X}-module of the form O_{F^{r},X} ⊗_{O_{X}} M for some O_{X}-module M, we refer to M as an induced left O_{F^{r},X}-module.

Note that if M = O_{F^{r},X} ⊗_{O_{X}} M is induced, then M/F M ⟶^{∼} M, and so M is quasi-coherent if and only if the same is true of M.

Lemma 1.10.1. Any quasi-coherent left O_{F^{r},X}-module is the quotient of a quasicoherent induced left O_{F^{r},X}-module. Thus any object of D_{qc}^{b}(O_{F^{r},X}) may be represented by a complex bounded above of quasi-coherent induced left O_{F^{r},X}-modules.

Proof. We already saw this in 1.7.1. □

(1.10.2) For any sheaf of O_{X}-modules M we have an isomorphism of functors on the category of O_{F^{r},X}-modules

Hom_{O_{X}} (M, –) ⟶^{∼} Hom_{O_{F^{r} ,X}} (O_{F^{r},X} ⊗_{O_{X}} M, –).

Thus if X is a smooth k-scheme then the forgetful functor from O_{F^{r},X}-modules to O_{X}-modules is right adjoint to the functor O_{F^{r},X} ⊗ –, which is shown to be exact by lemma 1.3.1. As a consequence we see that if I is an injective sheaf of O_{F^{r},X} modules, then I is injective as a sheaf of O_{X}-modules (provided of course that X is smooth over k).

(1.10.3) The previous paragraph provides a characterization of the morphisms between induced left O_{F^{r},X}-modules. Suppose that M and N are two O_{X}-modules. Using 1.10.2 and 1.3.2 we have the formula

Hom_{O_{F^{r} ,X}} (O_{F^{r},X} ⊗_{O_{X}} M, O_{F^{r},X} ⊗_{O_{X}} N)

⟶^{∼} Hom_{O_{X}} (M, O_{F^{r},X} ⊗_{O_{X}} N) ⟶^{∼} Hom_{O_{X}} (M, ⨁_{n=0}^{∞}(F_{X}^{rn})^{∗}N).

(1.11) We will need a variant of the induced module construction, which will yield bimodules for a certain pair of rings. We will need this construction in a relative setting, so we let f : Y → X be a morphism of smooth k-schemes. We put ourselves in the situation of the (r)^{th} relative Frobenius diagram of Y over X, as discussed in (A.2):

\xymatrix{Y \ar[r]^{F_{Y /X}^{(r)}} \ar[dr]^{f} & Y^{(r)} \ar[r]^{F_{X}^{r′}} \ar[d]^{f^{(r)}} & Y \ar[d]^{f} \\  & X \ar[r]^{F_{X}^{r}} & X.}

Recall that both F_{Y/X}^{(r)} and F_{X}^{r}′ induce homeomorphisms of underlying spaces, and their composition induces the identity on the underlying space of Y; thus we regard Y^{(r)} as having the same underlying space as Y, just a different scheme structure. In fact, O_{Y^{(r)}} = f^{−1}O_{X}^{(r)}⊗_{f^{−1}O_{X}} O_{Y} , and the morphisms F_{Y/X}^{(r)} and F_{X}^{r}′ correspond to the morphism O_{Y^{(r)}} → O_{Y} defined by a⊗b ↦ ab^{q}, and the morphism O_{Y} → O_{Y^{(r)}} defined by b ↦ 1 ⊗ b, respectively.

The sheaf-theoretic pull-back f^{−1}O_{F^{r},X} = f^{−1}O_{X}[F] is a sheaf of rings on Y, which naturally contains f^{−1}O_{X} as a sheaf of subrings. If M is a sheaf of O_{Y} - modules, then we may regard it as a sheaf of (f^{−1}O_{X}, O_{Y})-bimodules (where the f^{−1}O_{X} action is via the map f^{−1}O_{X} → O_{Y} corresponding to the morphism f), and so the tensor product f^{−1}O_{F^{r},X} ⊗_{f^{−1}O_{X}} M is naturally a sheaf of (f^{−1}O_{F^{r},X}, O_{Y})- bimodules.

Proposition-Definition 1.11.1. Let M be a sheaf of O_{Y} -modules equipped with a map ψ : F_{Y/X∗}^{(r)}M → F_{X}^{r′∗}M of O_{Y^{(r)}}-modules. Then the natural (f^{−1}O_{F^{r},X}, O_{Y})- bimodule structure on the tensor product f^{−1}O_{F^{r},X} ⊗_{f^{−1}O_{X}} M may be canonically extended to an (f^{−1}O_{F^{r},X}, O_{F^{r},Y})-bimodule structure, which we refer to as the bimodule structure induced by ψ.

Proof. Since F_{Y/X}^{(r)} is the identity on underlying topological spaces, we may identify

M and F_{Y/X∗}^{(r)}M as sheaves of abelian groups. Also

F_{X}^{r}′^{∗}M ⟶^{∼} f^{−1}O_{X}^{(r)} ⊗_{f^{−1}O_{X}} M = f^{−1}O_{X}F^{r} ⊗_{f^{−1}O_{X}} M

⊂ f^{−1}O_{X}[F^{r}] ⊗_{f^{−1}O_{X}} M = f^{−1}O_{F^{r},X} ⊗_{f^{−1}O_{X}} M.

Thus the morphism ψ yields a morphism of sheaves on Y,

σ : ω_{Y/X} → f^{−1}O_{F^{r},X} ⊗_{f^{−1}O_{X}} ω_{Y/X},

such that for any sections b of O_{Y} and η of ω_{Y/X},

σ(b^{q}η) = bσ(η).

We now define a right action of O_{F^{r},Y} on f^{−1}O_{F^{r},X} ⊗_{f^{−1}O_{X}} M as follows: the morphism induced by right multiplication by F^{r} is

f^{−1}O_{F^{r},X} ⊗_{f^{−1}O_{X}} M →^{id⊗σ} f^{−1}O_{F^{r},X} ⊗_{f^{−1}O_{X}} f^{−1}O_{F^{r},X} ⊗_{f^{−1}O_{X}} M

→ f^{−1}O_{F^{r},X} ⊗_{f^{−1}O_{X}} M,

where the second morphism is that induced by the ring structure of f^{−1}O_{F^{r},X}.

It is easily checked that this does indeed give f^{−1}O_{F^{r},X} ⊗_{f^{−1}O_{X}} M the structure of a right O_{F^{r},Y} -module which extends its O_{Y} -module structure. From the construction, and the associativity of multiplication in f^{−1}O_{F^{r},X}, it is immediate that f^{−1}O_{F^{r},X} ⊗_{f^{−1}O_{X}} M becomes an (f^{−1}O_{F^{r},X}, O_{F^{r},Y})-bimodule when endowed with a right O_{F^{r},Y} -module structure in this way. □

Lemma 1.11.2. Suppose that we have a pair of O_{Y} -modules M and M′ equippped with morphisms ψ : F_{Y/X∗}^{(r)}M → F_{X}^{r′∗}M and ψ′ : F_{Y/X∗}^{(r)}M′ → F_{X}^{r′∗}M′ of O_{Y^{(r)}}- modules, and a morphism of O_{Y} -modules θ : M → M′ such that the square

\xymatrix{F_{Y/X∗}^{(r)}M \ar[r]^{ψ} \ar[d]^{F_{Y /X∗}^{(r)}θ} & F_{X}^{r′∗}M \ar[d]^{F_{X}^{r′∗}θ} \\ F_{Y/X∗}^{(r)}M′ \ar[r]^{ψ} & F_{X}^{r′∗}M′}

commutes. Then the induced morphism

id ⊗ θ : f^{−1}O_{F^{r},X} ⊗_{f^{−1}O_{X}} M → f^{−1}O_{F^{r},X} ⊗_{f^{−1}O_{X}} M′

is a morphism of (f^{−1}O_{F^{r},X}, O_{F^{r},Y})-bimodules, when both source and target are equipped with their induced bimodule structures.

Proof. This is immediate from the construction. □

Remark 1.11.3 The map f^{#} : f^{−1}O_{X} → O_{Y} corresponding to the morphism f induces a map f^{−1}O_{X}[F^{r}] → O_{Y} [F^{r}]. Thus in the situation (1.11.2), we may restrict scalars on the right via this map of rings to obtain an (f^{−1}O_{F^{r},X}, f^{−1}O_{F^{r},X})- bimodule structure on f^{−1}O_{F^{r},X} ⊗_{f^{−1}O_{X}} M. We refer to the resulting structure as the induced (f^{−1}O_{F^{r},X}, f^{−1}O_{F^{r},X})-bimodule structure on f^{−1}O_{F^{r},X} ⊗_{f^{−1}O_{X}} M.

(1.11.4) We can apply the induced bimodule construction in the particular case when we have a single smooth k-scheme X, and f is the identity id_{X} : X → X. In this situation we find that if M is a sheaf of O_{X}-modules equipped with a morphism β : M → F_{X}^{r∗}M, the induced module O_{F^{r},X} ⊗_{O_{X}} M is naturally an (O_{F^{r},X}, O_{F^{r},X})-bimodule.

Note that in this situation the endomorphism of O_{F^{r},X} ⊗_{O_{X}} M (regarded as a left O_{F^{r},X}-module) corresponding to right multiplication by F^{r} is the map of induced modules corresponding to the morphism β : M → F_{X}^{r∗}M ⊂ ⨁_{n=0}^{∞} F_{X}^{rn∗}M via the discussion of (1.10.3).

(1.11.5) Let us now return to the relative setting of (1.11) above. There is a straightforward but important fact concerning this situation that we will need.

Suppose that M is an O_{Y} -module equipped with ψ : F_{Y/X∗}^{(r)}M → F_{X}^{r′∗}M. Ap-

plying f_{∗}^{(r)} yields a map f_{∗}^{(r)}ψ : f_{∗}F_{Y/X∗}^{(r)}M → f_{∗}^{(r)}F_{X}^{r′∗}M. Now functoriality of

push-forwards yields a natural isomorphism f_{∗}M ⟶^{∼} f_{∗}^{(r)}F_{Y/X∗}^{(r)}M, while flat base-

change applied to F_{X}^{r} yields an isomorphism F_{X}^{r∗}f_{∗}M ⟶^{∼} f_{∗}^{(r)}F_{X}^{r′∗}M. Via these isomorphisms we see that f_{∗}^{(r)}ψ induces a morphism f_{∗}M → F_{X}^{r∗}f_{∗}M, and thus O_{F^{r},X} ⊗_{O_{X}} f_{∗}M is equipped with an (O_{F^{r},X}, O_{F^{r},X})-bimodule structure.

On the other hand, we may equip the tensor product f^{−1}O_{F^{r},X} ⊗_{f^{−1}O_{X}} N with its induced (f^{−1}O_{F^{r},X}, O_{F^{r},Y})-bimodule structure, and underlying this is the induced (f^{−1}O_{F^{r},X}, f^{−1}O_{F^{r},X})-bimodule structure. Applying f_{∗}, we find that f_{∗}(f^{−1}O_{F^{r},X} ⊗_{f^{−1}O_{X}} N) is naturally equipped with an (O_{F^{r},X}, O_{F^{r},X})-bimodule structure. The projection formula applied to the locally free O_{X}-module O_{F^{r},X} yields a natural isomorphism O_{F^{r},X} ⊗_{O_{X}} f_{∗}M ⟶^{∼} f_{∗}(f^{−1}O_{F^{r},X} ⊗_{f^{−1}O_{X}} N). Thus the tensor product O_{F^{r},X} ⊗_{O_{X}} f_{∗}M is also equipped with an (O_{F^{r},X}, O_{F^{r},X})- bimodule structure via this construction.

Lemma 1.11.6. In the situation of (1.11.5), the two (O_{F^{r},X}, O_{F^{r},X})-bimodule structures defined on O_{F^{r},X} ⊗_{O_{X}} f_{∗}M are in fact equal.

Proof. This is simply a matter of chasing through definitions. □

(1.11.7) There is one final observation regarding induced bimodules that we wish to make. As in (1.11.4), we take f to be the identity map from a smooth k-scheme X to itself, and we consider the particular case when M is an O_{F^{r},X}-module whose structural morphism ϕ_{M} is an isomorphism. In this case, we may take β to be ϕ_{M^{•}}^{−1} : M → F_{X}^{∗}M. Then O_{F^{r},X} ⊗_{O_{X}} M is equipped with its induced bimodule structure. On the other hand, we may form the tensor product of left O_{F^{r},X}- modules M ⊗_{O_{X}} O_{F^{r},X}, and using the right multiplication of O_{F^{r},X} on itself, we see that this tensor product also has the natural structure of an (O_{F^{r},X}, O_{F^{r},X})- bimodule.

Lemma 1.11.8. In the situation of (1.11.7), there is a natural isomorphism of (O_{F^{r},X}, O_{F^{r},X})-bimodules

O_{F^{r},X} ⊗_{O_{X}} M ⟶^{∼} M ⊗_{O_{X}} O_{F^{r},X}.

Proof. Write O_{F^{r},X} ⊗_{O_{X}} M ⟶^{∼} ⨁_{n=0}^{∞} F_{X}^{rn∗}M (via corollary 1.3.2) and M ⊗_{O_{X}} O_{F^{r},X} ⟶^{∼} M[F^{r}] = ⨁_{n=0}^{∞} MF^{rn}. Define the isomorphism to be the direct sum of the isomorphisms

F_{X}^{rn∗}M ⟶^{{ϕ_{n,M} \atop ∼}} M ⟶^{{·F^{rn} \atop ∼}} MF^{rn}.

One easily checks that this is indeed a map of (O_{F^{r},X}, O_{F^{r},X})-bimodules. □

(1.12) It will be important in our later arguments to be able to make a change of ground field in the theory of O_{F^{r}}-modules, and so we now explain this. Let k′ be an extension field of k. If X is a smooth k-scheme, then X′ = k′ ⊗_{k} X is a smooth k′-scheme. If M is an O_{F^{r},X}-module, then k′ ⊗_{k} M is naturally an O_{F^{r},X′}-module (with F^{r} acting via a⊗m ↦ a^{q} ⊗F^{r} ·m). This restricts to a functor k′ ⊗_{k} – : μ(X) → μ(X′). The functor k′ ⊗_{k} – is exact (since k′ is flat over k) and so induces conservative functors (which hence reflect isomorphisms, since a map in a triangulated category is an isomorphism exactly when its cone is zero)

k′ ⊗_{k} – : D^{•}(O_{F^{r},X}) → D^{•}(O_{F^{r},X′})

(where • is any of b, +, −, or ∅) and

k′ ⊗_{k} – : D_{qc}^{b}(O_{F^{r},X}) → D_{qc}^{b}(O_{F^{r},X′}).

Note that this last functor interchanges (in an obvious sense) – ⊗^{𝕃}_{O_{X}} – and – ⊗^{𝕃}_{O′_{X}} –.

(1.13) If X is a k-scheme, we let D^{•}(X) denote the derived category of complexes of 𝔽_{q}-sheaves on X; here • can be any one of +, −, b or ∅. If M^{•} and N^{•} are two objects in D^{b}(O_{F^{r},X}), then we may form \underline{RHom}_{O_{F^{r} ,X}}^{•} (M^{•}, N^{•}), which is an object of D^{+}(X). (Note that the constant sheaf 𝔽_{q} is contained in the centre of O_{F^{r},X}.)

Such sheaves will be at the basis of our later definition of the Riemann-Hilbert correspondence, so their functoriality is of great interest to us.

Lemma 1.13.1. Suppose that M_{1}^{•}, M_{2}^{•}, N_{1}^{•} and N_{2}^{•} are objects of D_{qc}^{b}(O_{F^{r},X}). Then there is a natural transformation

\underline{RHom}_{O_{F^{r} ,X}}^{•} (M_{1}^{•}, N_{1}^{•}) ⊗_{𝔽_{q}} \underline{RHom}_{O_{F^{r} ,X}}^{•} (M_{2}^{•}, N_{2}^{•})

⟶ \underline{RHom}_{O_{F^{r} ,X}}^{•} (M_{1}^{•} ⊗^{𝕃}_{O_{X}} M_{2}^{•}, N_{1}^{•} ⊗^{𝕃}_{O_{X}} N_{2}^{•}).

This natural transformation is compatible with change of ground field.

Proof. Using lemmas 1.7.1 and 1.7.2 we may assume that both M_{1}^{•} and M_{2}^{•} are bounded above complexes of free O_{F^{r},X}-modules, and that N_{1}^{•} and N_{2}^{•} are bounded complexes of O_{F^{r},X}-modules which are O_{X}-flat.

Then we have

\underline{RHom}_{O_{F^{r} ,X}}^{•} (M_{1}^{•}, N_{1}^{•}) ⊗_{𝔽_{q}} \underline{RHom}_{O_{F^{r} ,X}}^{•} (M_{2}^{•}, N_{2}^{•})
=\underline{Hom}_{O_{F^{r} ,X}}^{•} (M_{1}^{•}, N_{1}^{•}) ⊗_{𝔽_{q}} \underline{Hom}_{O_{F^{r} ,X}}^{•} (M_{2}^{•}, N_{2}^{•})
⟶\underline{Hom}_{O_{F^{r} ,X}}^{•} (M_{1}^{•} ⊗_{O_{X}} M_{2}^{•}, N_{1}^{•} ⊗_{O_{X}} N_{2}^{•})
=\underline{RHom}_{O_{F^{r} ,X}}^{•} (M_{1}^{•} ⊗^{𝕃}_{O_{X}} M_{2}^{•}, N_{1}^{•} ⊗^{𝕃}_{O_{X}} N_{2}^{•}).

This is the required transformation, and is clearly compatible with change of ground field. □

Lemma 1.13.2. Let k′ be an extension field of k, let X be a smooth k-scheme, let X′ denote the base-change of X over k′, and let p_{X} : X′ → X denote the natural morphism. Then for any two objects M^{•} and N^{•} of D^{b}(O_{F^{x},X}), there is a natural morphism

p_{X}^{−1}\underline{RHom}_{O_{F^{r} ,X}}^{•} (M^{•}, N^{•}) → \underline{RHom}_{O_{F^{r} ,X′}}^{•} (k′ ⊗_{k} M^{•}, k′ ⊗_{k} N^{•}).

Proof. Let M^{•} → I^{•} be a quasi-isomorphism, with I^{•} a bounded below complex of injective O_{F^{r},X}-modules. Since k′ ⊗_{k} – is exact, k′ ⊗_{k} M^{•} → k′ ⊗_{k} I^{•} is again a quasi-isomorphism. Let k′ ⊗_{k} I^{•} → I^{′•} be a quasi-isomorphism with I^{′•} a bounded below complex of injective O_{F^{r},X′}-modules. Then we have

p_{X}^{−1}\underline{RHom}_{O_{F^{r} ,X}}^{•} (M^{•}, N^{•}) = p_{X}^{−1}\underline{Hom}_{O_{F^{r} ,X}}^{•} (M^{•}, I^{•})

⟶ \underline{Hom}_{O_{F^{r} ,X′}}^{•} (k′ ⊗_{k} M^{•}, k′ ⊗_{k} I^{•})

⟶ \underline{Hom}_{O_{F^{r} ,X′}}^{•} (k′ ⊗_{k} M^{•}, I^{′•})

⟶^{∼} \underline{RHom}_{O_{F^{r} ,X′}}^{•} (k′ ⊗_{k} M^{•}, I^{′•}).

This is the required natural transformation. □

Remark 1.13.3 If M^{•} is a complex in D_{qc}^{b}(O_{F^{r},X}), then we can compute the natural transformation of the preceding lemma by replacing M^{•} by a bounded above locally free resolution, rather than by replacing N^{•} by a bounded below injective resolution. This will prove useful in the proof of proposition 6.8 below.

(1.13.4) Taking global sections of the morphism of lemma 1.13.2 induces a morphism

RHom_{O_{F^{r} ,X}} (M^{•}, N^{•}) → RHom_{O_{F^{r} ,X′}} (k′ ⊗_{k} M^{•}, k′ ⊗_{k} N^{•}).

It is clear from the construction that the corresponding morphism

Hom_{O_{F^{r} ,X}} (M^{•}, N^{•}[i]) → Hom_{O_{F^{r} ,X′}} (k′ ⊗_{k} M^{•}, k′ ⊗_{k} N^{•}[i])

induced between the i^{th} cohomology groups of these complexes is just the morphism of Hom’s induced by the functor k′ ⊗_{k} –.

(1.13.5) One particular case of lemma 1.13.2 that will concern us later is when k′ is a purely inseparable extension of k. In this situation, the morphism p_{X} of the lemma induces a homeomorphism of the underlying topological spaces. If we use this to identify X′ with X as topological spaces, the natural transformation of the lemma reduces to a morphism

\underline{RHom}_{O_{F^{r} ,X}}^{•} (M^{•}, N^{•}) → \underline{RHom}_{O_{F^{r} ,X′}}^{•} (k′ ⊗_{k} M^{•}, k′ ⊗_{k} N^{•}).

2. pull-backs of O_{F^{r}}-modules.

(2.1) Suppose that f : Y → X is a morphism of smooth k-schemes. We will define a morphism of triangulated categories

f^{!} : D_{qc}^{b}(O_{F^{r},X}) → D_{qc}^{b}(O_{F^{r},Y})

called “pull-back by f.” (Note that if M is an O_{F^{r},X}-module which is quasicoherent as an O_{X}-module, then the underlying O_{Y} module f^{!}M does not in general coincide with the O_{Y} -module which is usually denoted by f^{!}M in the duality theory of quasi-coherent sheaves [Ha]. In fact, temporarily denoting the latter by ‘f^{!}’, on the level of O_{Y} -modules there will be isomorphisms

f^{!}M^{•} ⟶^{∼} 𝕃f^{∗}M^{•}[d_{Y/X}] ⟶^{∼} ω_{Y/X}^{−1} ⊗_{O_{Y}} ‘f^{!}’M^{•}.

In the following f^{!} will denote the O_{F^{r}}-module pull-back that we define in this section, except when we explicitly state otherwise.)

(2.2) Let f^{#} : f^{−1}O_{X} → O_{Y} denote the morphism of sheaves of rings on Y arising from the morphism of schemes f. Then (as was already observed in remark 1.11.3) f^{#} induces a morphism of sheaves of rings

f^{−1}O_{F^{r},X} = f^{−1}O_{X}[F^{r}] → O_{Y} [F^{r}] = O_{F^{r},Y} .

We use this morphism to make O_{F^{r},Y} a right f^{−1}O_{F^{r},X}-module.

Definition 2.2.1. O_{F^{r},Y} can be regarded as an (O_{F^{r},Y} , f^{−1}O_{F^{r},X})-bimodule, via its standard left O_{F^{r},Y} -module structure, and the right f^{−1}O_{F^{r},X}-structure constructed above. We denote this bimodule by O_{F^{r},Y →X}.

(2.3) We now define the pull-back functor.

Definition 2.3.1. Let f : Y → X be a morphism of smooth k-schemes. We define f^{!} : D_{qc}^{b}(O_{F^{r},X}) → D_{qc}^{b}(O_{F^{r},Y}) as follows:

f^{!}M^{•} = O_{F^{r},Y →X} ⊗^{𝕃}_{f^{−1}O_{F^{r} ,X}} f^{−1}M^{•}[d_{Y/X}].

In order for this definition to make sense, we must show both that we can com-

pute the ⊗^{𝕃}, and that the resulting complex does indeed lie in D_{qc}^{b}(O_{F^{r},X}). That the

⊗^{𝕃} is well-defined follows from lemma 1.7.1, as f^{−1} is exact. However, it remains to show that the resulting complex has only finitely many non-zero cohomology sheaves, and that these are quasi-coherent. That this is so is proved in the following lemma.

Lemma 2.3.2. If M^{•} is a complex in D_{qc}^{b}(O_{F^{r},X}), then f^{!}(M^{•}) has only finitely many non-zero cohomology sheaves, and defines an object of D_{qc}^{b}(O_{F^{r},Y}).

Proof. If M^{•} is an object of D_{qc}^{b}(O_{F^{r},X}), corollary 1.7.2 shows that M^{•} has a finite length resolution P^{•} by quasi-coherent left O_{F^{r},X}-modules which are flat as O_{X}-modules, so that we have

f^{!}M^{•} ⟶^{∼} O_{F^{r},Y →X} ⊗^{𝕃}_{f^{−1}O_{F^{r} ,X}} M^{•}[d_{Y/X}]

⟶^{∼} O_{Y} ⊗^{𝕃}_{f^{−1}O_{X}} M^{•}[d_{Y/X}] ⟶^{∼} O_{Y} ⊗_{f^{−1}O_{X}} P^{•}[d_{Y/X}]

is a bounded length complex of quasi-coherent left O_{F^{r},Y} -modules. (Note that, for example, F^{r} acts diagonally on O_{Y} ⊗_{f^{−1}O_{X}} P^{•} (cf (1.9)).) □

Proposition 2.4. Let f : Y → X and g : Z → Y be two morphisms of smooth k-schemes. Then the functors g^{!}f^{!} and (fg)^{!} from D_{qc}^{b}(O_{F^{r},X}) to D_{qc}^{b}(O_{F^{r},Z}) are naturally isomorphic.

Proof. We let M^{•} be a complex of D_{qc}^{b}(O_{F^{r},X}). It is quasi-isomorphic to a bounded complex whose members are flat as O_{X}-modules (see the proof of the preceding lemma) and so we replace it by such a complex, which we continue to denote by M^{•}.

Then f^{!}M^{•} = O_{Y} ⊗_{f^{−1}O_{X}} f^{−1}M^{•}[d_{Y/X}], and the right side of this equation is a bounded complex of O_{Y,F} -modules whose members are flat as O_{Y} -modules. Thus we may use the right hand side to compute g^{!}f^{!}M^{•}, and we see that

g^{!}f^{!}M^{•} ⟶^{∼}O_{Z} ⊗_{g^{−1}O_{Y}} g^{−1}(O_{Y} ⊗_{f^{−1}O_{X}} f^{−1}M^{•}[d_{Y/X}])[d_{Z/Y}]

⟶^{∼}O_{Z}⊗_{g^{−1}O_{Y}} g^{−1}O_{Y} ⊗_{g^{−1}f^{−1}O_{X}} g^{−1}f^{−1}M^{•}[d_{Z}] ⟶^{∼}O_{Z}⊗_{g^{−1}f^{−1}O_{X}} g^{−1}f^{−1}M^{•}[d_{Z}]

⟶^{∼} O_{Z} ⊗_{(fg)^{−1}O_{X}} (fg)^{−1}M^{•}[d_{Z}] ⟶^{∼} (fg)^{!}M^{•}.

□

Proposition 2.5. Let f : Y → X be a morphism of smooth k-schemes. If M^{•} and N^{•} are objects of D_{qc}^{b}(O_{F^{r},X}) then there is a natural isomorphism

f^{!}(M^{•} ⊗^{𝕃}_{O_{X}} N^{•}) → f^{!}M^{•} ⊗^{𝕃}_{O_{Y}} f^{!}N^{•}[d_{X/Y}].

Proof. We may suppose that M^{•} and N^{•} are both finite-length complexes of O_{X}- flat modules. Then we compute

f^{!}(M^{•} ⊗^{𝕃}_{O_{X}} N^{•}) = f^{!}(M^{•} ⊗_{O_{X}} N^{•}) = O_{Y} ⊗_{f^{−1}O_{X}} f^{−1}(M^{•} ⊗_{O_{X}} N^{•})[d_{Y/X}]

⟶^{∼} (O_{Y} ⊗_{f^{−1}O_{X}} f^{−1}M^{•}[d_{Y/X}]) ⊗_{O_{Y}} (O_{Y} ⊗_{f^{−1}O_{X}} f^{−1}M^{•}[d_{Y/X}])[d_{X/Y}]

= f^{!}M^{•} ⊗^{𝕃}_{O_{Y}} f^{!}N^{•}.

This proves the proposition. □

Proposition 2.6. Let f : Y → X be a morphism of smooth k-schemes. If M^{•} and N^{•} are two objects of D_{qc}^{b}(O_{F^{r},X}), then there is a natural transformation

f^{!}\underline{RHom}_{O_{F^{r} ,X}}^{•} (M^{•}, N^{•}) → \underline{RHom}_{O_{F^{r} ,Y}}^{•} (f^{!}M^{•}, f^{!}N^{•}).

If f is an open immersion, this natural transformation is in fact an isomorphism.

Proof. We may assume that M^{•} is a be a bounded above complex of locally free left O_{F^{r},X}-modules, and that N^{•} is a finite length complex of left O_{F^{r},X}-modules which are flat as O_{X}-modules. Then we obtain the natural transformation

f^{−1}\underline{RHom}_{O_{F^{r} ,X}}^{•} (M^{•}, N^{•})

= f^{−1}\underline{Hom}_{O_{F^{r} ,X}}^{•} (M^{•}, N^{•}) ⟶ \underline{Hom}_{f^{−1}O_{F^{r} ,X}}^{•} (f^{−1}M^{•}, f^{−1}N^{•})

⟶ \underline{Hom}_{O_{F^{r} ,Y}}^{•} (O_{F^{r},Y →X} ⊗_{f^{−1}O_{F^{r} ,X}} f^{−1}M^{•}, O_{F^{r},Y →X} ⊗_{f^{−1}O_{F^{r} ,X}} f^{−1}N^{•})

⟶^{∼} \underline{RHom}_{O_{F^{r} ,Y}}^{•} (f^{!}M^{•}[d_{X/Y}], f^{!}N^{•}[d_{X/Y}]) ⟶^{∼} \underline{RHom}_{O_{F^{r} ,Y}}^{•} (f^{!}M^{•}, N^{•}),

where the second-to-last isomorphism follows because each of M^{•} and N^{•} is a complex of O_{X}-flat modules and O_{F^{r},Y →X} ⊗_{f^{−1}O_{F^{r} ,X}} f^{−1}M^{•} is furthermore a locally free complex of O_{F^{r},Y} -modules. If f is an open immersion, then it is immediate that all the arrows is this composite are isomorphisms. □

Remark 2.6.1 Taking global sections, the morphism of proposition 2.6 yields a morphism

RHom^{•}_{O_{F^{r} ,X}} (M^{•}, N^{•}) → RHom^{•}_{O_{F^{r} ,Y}} (f^{!}M^{•}, f^{!}N^{•}).

One sees from the construction that the corresponding morphism

Hom_{D^{b}(O_{F^{r} ,X})}(M^{•}, N^{•}[i]) → Hom_{D^{b}(O_{F^{r} ,Y} )}(f^{!}M^{•}, f^{!}N^{•}[i])

induced between the i^{th} cohomology groups of these complexes is just the morphism of Hom’s induced by the functor f^{!}.

Proposition 2.7. Let f : Y → X be a morphism of smooth k-schemes, let k′/k be a field extension, and let f′ : Y′ → X′ be the base-change of f over k. Then the diagram

\xymatrix{D_{qc}^{b}(O_{F^{r},X}) \ar[d]^{f^{!}} \ar[r]^{k′⊗_{k}–} & D_{qc}^{b}(O_{F^{r},X′}) \ar[d]^{f^{′!}} \\ D_{qc}^{b}(O_{F^{r},Y}) \ar[r]^{k′⊗_{k}–} & D_{qc}^{b}(O_{F^{r},Y′})}

commutes up to natural isomorphism, in a manner compatible with the natural isomorphisms of propositions 2.4, 2.5 and 2.6.

Proof. This is a consequence of the standard compatibilities satisfied by tensor products. □

(2.8) In this subsection we study the pull-back by a closed immersion f : Y → X of smooth k-schemes in more detail. We put ourselves in the context of the r^{th} Frobenius diagram of f. (See the appendix for a discussion of this diagram, as well as a summary of and a discussion of our conventions concerning duality of quasi-coherent sheaves, which will be used below.)

Lemma 2.8.1. If N is a sheaf of O_{Y} -modules equipped with a morphism

(F_{X}^{r′∗}N)[I] → N,

then ω_{Y/X}^{−1} ⊗_{O_{Y}} N is naturally equipped with the structure of a left O_{F^{r},Y} -module.

Proof. It is explained in (A.2.2) that the relative Cartier operator C_{Y/X}^{(r)} induces an isomorphism

ω_{Y/X} ⊗_{O_{Y}} F_{Y}^{r∗}(ω_{Y/X}^{−1} ⊗_{O_{Y}} N) ⟶^{∼} (F_{X}^{r′∗}(ω_{Y/X} ⊗_{O_{Y}} ω_{Y/X}^{−1} ⊗_{O_{Y}} N))[I] ⟶^{∼} (F_{X}^{r′∗}N)[I].

Composing with the given morphism (F_{X}^{r′∗}N)[I] → N and then tensoring on the left by ω_{Y/X}^{−1}, we obtain a morphism F_{Y}^{∗} (ω_{Y/X}^{−1} ⊗_{O_{Y}} N) → ω_{Y/X}^{−1} ⊗_{O_{Y}} N, which by lemma 1.5.2 is the structural isomorphism of a unique left O_{F^{r},Y} -module structure on ω_{Y/X}^{−1} ⊗_{O_{Y}} N. □

Corollary 2.8.2. If M is a left O_{F^{r},X}-module then ω_{Y/X}^{−1} ⊗_{O_{Y}} M[I] is naturally equipped with the structure of a left O_{F^{r},Y} -module.

Proof. Base-change via the flat map F_{X}^{r} yields a natural isomorphism

F_{X}^{r′∗}(M[I]) ⟶^{∼} (F_{X}^{r∗}M)[I^{(r)}].

The structural morphism F_{X}^{r∗}M → M yields a map (F_{X}^{r∗}M)[I^{(r)}] → M[I^{(r)}]. Composing these two, we get a map F_{X}^{r′∗}(M[I]) → M[I^{(r)}], and hence a map (F_{X}^{r′∗}(M[I]))[I] → (M[I^{(r)}])[I]. The previous lemma now applies. □

(2.8.3) Since O_{F^{r},X} is flat as a right O_{X}-module, any injective left O_{F^{r},X}- module is also an injective O_{X}-module. Thus, by computing with resolutions by injective left O_{F^{r},X}-modules, we may use corollary 2.8.2 to define the derived functor

ω_{Y/X}^{−1} ⊗_{O_{Y}} \underline{RHom}_{O_{X}}^{•} (f_{∗}O_{Y} , –) : D^{+}(O_{F^{r},X}) → D^{+}(O_{F^{r},Y}).

Since \underline{RHom}_{O_{X}}^{•} (f_{∗}O_{Y} , –) has cohomological dimension d_{Y/X} (< ∞), as one sees by computing locally with Koszul complexes, and since M ↦ M[I] takes quasicoherent O_{X}-modules to quasi-coherent O_{Y} -modules, we see that this functor restricts to a functor

ω_{Y/X}^{−1} ⊗_{O_{Y}} \underline{RHom}_{O_{X}}^{•} (f_{∗}O_{Y} , –) : D_{qc}^{b}(O_{F^{r},X}) → D_{qc}^{b}(O_{F^{r},Y}).

Proposition 2.8.4. There is a natural isomorphism of functors

ω_{Y/X}^{−1} ⊗_{O_{Y}} \underline{RHom}_{O_{X}}^{•} (f_{∗}O_{Y} , –) ⟶^{∼} f^{!}.

On the level of O_{Y} -modules this is the (twist by ω_{Y/X}^{−1} of the) isomorphism (A.1.4).

Proof. It suffices to construct a natural isomorphism

ω_{Y/X}^{−1} ⊗_{O_{Y}} \underline{Ext}_{O_{X}}^{d_{X/Y}} (f_{∗}O_{Y} , –) ⟶^{∼} H^{d_{X/Y}} (f^{!}).  (2.8.5)

The desired isomorphism will then follow by homological algebra ([Ha, I 7.4] and [Con, I 2.1]).

There is an isomorphism on the level of O_{Y} -modules H^{d_{X/Y}} (f^{!}) ⟶^{∼} f^{∗}, and the fundamental local isomorphism (see the discussion and references in (A.1.3)) yields (2.8.5) on the level of O_{Y} -modules. We will show that this is compatible with the O_{F^{r},Y} -module structures on each side. This involes unwinding all the relevant constructions.

One thing that makes this slightly awkward is that the the functor F_{Y/X}^{!}, which is the derived functor of the functor M ↦ M[I] on O_{Y^{(r)}}-modules appearing in the statement of lemma 2.8.1 and the construction of corollary 2.8.2, usually has infinite cohomological dimension. This makes the edge morphism

\underline{Ext}_{O_{X}}^{d_{X/Y}} (f_{∗}O_{Y} , –) → \underline{Ext}_{O_{X}}^{d_{X/Y ^{(r)}}} (f_{∗}^{(r)}O_{Y^{(r)}}, –)

discussed in (A.2.2) a little difficult to analyze.

We can get around this problem by the following device. If M is flat as a left O_{F^{r},X}-module (and so also flat as an O_{X}-module) then

\underline{Ext}_{O_{X}}^{i} (f_{∗}O_{Y} , M) = \underline{Ext}_{O_{X}}^{i} (f_{∗}^{(r)}O_{Y^{(r)}}, M) = 0

if i ≠ d_{Y/X}, and so the above edge map in this case becomes an isomorphism

(2.8.6)  \underline{Ext}_{O_{X}}^{d_{X/Y}} (f_{∗}O_{Y} , M) ⟶^{∼} \underline{Ext}_{O_{X}}^{d_{X/Y ^{(r)}}} (f_{∗}^{(r)}O_{Y^{(r)}}, M)[I].

Since any left O_{F^{r},X}-module may be written as the quotient of a flat left O_{F^{r},X}- module (via an obvious analogue of [Ha, II 1.2]), and since the functors appearing on either side of (2.8.5) are left exact functors of O_{F^{r},X}-modules, it suffices to verify that the isomorphism of (2.8.5) respects the O_{F^{r},Y} -module structures on each side in the case that M is a flat left O_{F^{r},X}-module. We assume that M is of this form from now on.

Let a_{1}, ⋯, a_{s} be a regular sequence which (locally) generates I. We have the following commutative diagram, in which the horizontal arrows are provided by the fundamental local isomorphism, and in which the vertical arrows are constructed using base-change by the flat isomorphism F_{X}^{r}, the structural morphism F_{X}^{∗}M → M, and the inverse of the isomorphism (2.8.6) (the explicit descriptions of the right hand arrows being explained in (A.2.2); in particular, the morphism labelled (1) is given by the formula a_{1}^{q−1} ⋯ a_{s}^{q−1}(da_{1}^{q} ∧⋯∧da_{s}^{q})^{−1} ⊗m ↦ (da_{1} ∧⋯∧da_{s})^{−1} ⊗m):

[figure: (F_{X}^{r′∗}\underline{Ext}_{O_{X}}^{d_{X/Y}} (f_{∗}O_{Y} , M))[I]; ∼; ∼; (F_{X}^{r′∗}(ω_{Y/X} ⊗_{O_{Y}} f^{∗}M))[I]; ∼; (F_{X}^{r′∗}ω_{Y/X} ⊗_{O_{Y ^{(r)}}} F_{X}^{r′∗}M)[I]; ∼; \underline{Ext}_{O_{X}}^{d_{X/Y ^{(r)}}} (f_{∗}^{(r)}O_{Y^{(r)}}, F_{X}^{r∗}M))[I]; ∼; (ω_{Y^{(r)}/X} ⊗_{O_{Y ^{(r)}}} f^{(r)∗}F_{X}^{r∗}M)[I]; \underline{Ext}_{O_{X}}^{d_{X/Y ^{(r)}}} (f_{∗}^{(r)}O_{Y^{(r)}}, M))[I]; ∼; ∼; (ω_{Y^{(r)}/X} ⊗_{O_{Y ^{(r)}}} f^{(r)∗}M)[I]; a_{1}^{q−1} ⋯ a_{s}^{q−1}ω_{Y^{(r)}/X} ⊗_{O_{Y ^{(r)}}} f^{(r)∗}M; (1); \underline{Ext}_{O_{X}}^{d_{X/Y}} (f_{∗}O_{Y} , M); ∼; ω_{Y/X} ⊗_{O_{Y}} f^{∗}M.]

If one applies the construction of lemma 2.8.1 to the composite of the left hand arrows in this diagram, one obtains the O_{F^{r},Y} -module structure defined on the left hand side of (2.8.5).

On the other hand, recalling from (A.2.2) the explicit formula for the relative Cartier operator

C_{Y/X}^{(r)} : (da_{1}^{q} ∧ ⋯ ∧ da_{s}^{q})^{−1} → a_{1}^{q−1} ⋯ a_{s}^{q−1}(da_{1}^{q} ∧ ⋯ ∧ da_{s}^{q})^{−1},

we find that applying the construction of lemma 2.8.1 to the composite of the right hand arrows of this diagram yields the structural morphism

F_{Y}^{∗} f^{∗}M ⟶^{∼} f^{∗}F_{X}^{∗}M ⟶^{f^{∗}ϕ_{M}} f^{∗}M,

which is exactly the structural morphism of H^{d_{Y /X}} (f^{!}M). Thus (2.8.5) is an isomorphism of O_{F^{r},Y} -modules, and the proof of the proposition is complete. □

Remark 2.8.7 An alternative description of the isomorphism of proposition 2.8.4, via residual complexes, is given in the course of proving proposition 4.5.3.

(2.9) If X is a smooth k-scheme, then we may consider the r^{th} relative Frobenius diagram of X over k (see A.2)

\xymatrix{X \ar[r]^{F_{X/k}^{(r)}} \ar[dr] & X^{(r)} \ar[r]^{F_{k}^{r′}} \ar[d] & X \ar[d] \\  & Spec k \ar[r]^{F_{k}^{r}} & Spec k.}

In this diagram, F_{X/k}^{(r)} is a finite flat morphism of smooth k-schemes. Thus

F_{X/k}^{(r)!} : D_{qc}^{b}(O_{F^{r},X^{(r)}}) → D_{qc}^{b}(O_{F^{r},X})

is simply pull-back via F_{X/k}^{(r)} on the level of O_{X^{(r)}}-modules, and if we compose this with the functor from D_{qc}^{b}(O_{F^{r},X}) to D_{qc}^{b}(O_{F^{r},X^{(r)}}) defined as base-change via the q^{th}-power map from k to itself, we obtain a functor which we denote

F_{X}^{r∗} : D_{qc}^{b}(O_{F^{r},X}) → D_{qc}^{b}(O_{F^{r},X}),

which on the level of O_{X}-modules is just the usual pull-back by F_{X}^{r}. (We are forced to describe this functor in this slightly round-about manner, because by restricting ourselves to the context of k-schemes and morphisms of k-schemes, we have not defined a pull-back functor for morphisms such as F_{X}^{r} which are not k-linear.)

If M^{•} is a complex in D_{qc}^{b}(O_{F^{r},X}), then the structural morphisms of each member of M^{•} yield a morphism

ϕ_{M^{•}} : F_{X}^{r∗}M^{•} → M^{•}

in D_{qc}^{b}(O_{F^{r},X}), which we refer to as the structural morphism of M^{•}. Note that since F_{X}^{r} is a flat morphism, formation of the structural morphism commutes with the passage to cohomology, in the sense that we have a commuative diagram of natural transformations

(2.9.1)  \xymatrix{F_{X}^{r∗}H^{i}(M^{•}) \ar[d]^{∼} \ar[r]^{ϕ_{H^{i}(M^{•})}} & H^{i}(M^{•}) \\ H^{i}(F_{X}^{r∗}M^{•}). \ar[ur]_{H^{i}(ϕ_{M^{•}})}}

Let us remark that since all the tensor products considred in the definition of the functor F_{X}^{r∗} on D_{qc}^{b}(O_{F^{r},X}) are taken with flat objects, this functor extends to a functor on D^{•}(O_{F^{r},X}), with • being any one of +, −, b or ∅, and (2.9.1) remains valid in this context.

(2.9.2) Let M^{•} and N^{•} be two complexes in D_{qc}^{b}(O_{F^{r},X}). Then we get a commutative diagram of complexes of sheaves on X:

\xymatrix{\underline{RHom}_{O_{F^{r} ,X}}^{•} (M^{•}, F_{X}^{r∗}N^{•}) \ar[r] \ar[d] & \underline{RHom}_{O_{F^{r} ,X}}^{•} (F_{X}^{r∗}M^{•}, F_{X}^{r∗}N^{•}) \ar[d] \\ \underline{RHom}_{O_{F^{r} ,X}}^{•} (M^{•}, N^{•}) \ar[r] \ar[ur] & \underline{RHom}_{O_{F^{r} ,X}}^{•} (F_{X}^{r∗}M^{•}, N^{•}),}

in which the horizontal arrows are induced by ϕ_{M^{•}} : F_{X}^{r∗}M^{•} → M^{•}, the vertical arrows are induced by ϕ_{M^{•}} : F_{X}^{r∗}N^{•} → N^{•}, and the diagonal arrow is induced by the functor F_{X}^{r∗} (by combining lemma 1.13.2, remark 1.13.5 and proposition 2.6).

(2.9.3) Suppose that k′ is a field extension of k, and let X′ (respectively X^{(r)′}) denote the base-change of X (respectively X^{(r)}) over k′. Then X^{(r)′} is naturally identified with X^{′(r)} (the base-change of X′ over k′ via the r^{th} power of the Frobenius endomorphism of k′), and under this identification, F_{X′/k′} is identified with the base-change of F_{X/k}. Thus for any complex M^{•} in D_{qc}^{b}(O_{F^{r},X}) there is a natural isomorphism

k′ ⊗_{k} F_{X}^{r∗}M^{•} ⟶^{∼} F_{X′}^{r∗}(k′ ⊗_{k} M^{•}).

As in lemma 1.13.2, let p_{X} denote the natural map X′ → X. Then for any pair of complexes M^{•}, N^{•} in D_{qc}^{b}(O_{F^{r},X}) there is a commutative diagram (in which we have written (–)_{k′} to denote k′ ⊗_{k} –)

\xymatrix{p_{X}^{−1}\underline{RHom}_{O_{F^{r} ,X}}^{•} (M^{•}, N^{•}) \ar[r]^{p_{X}^{−1}F_{X}^{r∗}} \ar[dd]^{k′⊗_{k}–} & p_{X}^{−1}\underline{RHom}_{O_{F^{r} ,X}}^{•} (F_{X}^{r∗}M^{•}, F_{X}^{r∗}N^{•}) \ar[d]^{k′⊗_{k}–} \\  & p_{X}^{−1}\underline{RHom}_{O_{F^{r} ,X′}}^{•} ((F_{X}^{r∗}M^{•})_{k′}, (F_{X}^{r∗}N^{•})_{k′}) \ar[d]^{∼} \\ \underline{RHom}_{O_{F^{r} ,X′}}^{•} (M_{k′}^{•}, N_{k′}^{•}) \ar[r]^{F_{X′}^{r∗}} & \underline{RHom}_{O_{F^{r} ,X′}}^{•} (F_{X′}^{r∗}(M_{k′}^{•}), F_{X′}^{r∗}(N_{k′}^{•}))}

(in which the vertical morphisms are those provided by lemma 1.13.2).

Recall from remark 1.13.5 that if k′ is purely inseparable algebraic extension of k then we may use p_{X} to identify X′ and X as topological spaces, and that having done this, we omit p_{X}^{−1} from the notation. The following lemma studies a particular case of this situation.

Lemma 2.9.4. Let k′ be a (necessarily purely inseparable extension) of k such that (k′)^{q} is contained in k. Then for any two complexes in D_{qc}^{b}(O_{F^{r},X}) there is a natural morphism

\underline{RHom}_{O_{F^{r} ,X′}}^{•} (M_{k′}^{•}, N_{k′}^{•}) → \underline{RHom}_{O_{F^{r} ,X}}^{•} (F_{X}^{r∗}M^{•}, F_{X}^{r∗}N^{•})

such that the diagram

\xymatrix{\underline{RHom}_{O_{F^{r} ,X}}^{•} (M^{•}, N^{•}) \ar[r] \ar[d] & \underline{RHom}_{O_{F^{r} ,X}}^{•} (F_{X}^{r∗}M^{•}, F_{X}^{r∗}N^{•}) \ar[d] \\ \underline{RHom}_{O_{F^{r} ,X′}}^{•} (M_{k′}^{•}, N_{k′}^{•}) \ar[r] \ar@{-->}[ur] & \underline{RHom}_{O_{F^{r} ,X′}}^{•} (F_{X′}^{r∗}(M_{k′}^{•}), F_{X′}^{r∗}(N_{k′}^{•}))}

(in which the outer edges of the diagram are provided by the discussion of (2.9.3)) commutes.

Proof. Since k^{q} is contained in k, we have a sequence of morphisms k ⟶ k′ ⟶^{a↦a^{q}} k. Considering the corresponding maps on Spec’s allows us to factor the endomorphism F_{k}^{r} of Spec k as a composite Spec k → Spec k′ → Spec k. Since F_{X}^{r∗} is computed by first base-changing via the Frobenius endomorphism of k, and then pulling-back via F_{X/k}^{(r)}, we see that it may equally well computed by first base-changing from k′, then base-changing from k′ to k (via the q^{th} power map), and then pulling-back via F_{X/k}^{(r)}. Thus (combining lemma 1.13.2, remark 1.13.5 and proposition 2.6) we get a commutative triangle

\xymatrix{\underline{RHom}_{O_{F^{r} ,X}}^{•} (M^{•}, N^{•}) \ar[r] \ar[d] & \underline{RHom}_{O_{F^{r} ,X}}^{•} (F_{X}^{r∗}M^{•}, F_{X}^{r∗}N^{•}) \\ \underline{RHom}_{O_{F^{r} ,X′}}^{•} (M_{k′}^{•}, N_{k′}^{•}) \ar@{-->}[ur]}

whose diagonal arrow is the desired map. This triangle is exactly the upper portion of the diagram appearing in the statement of the lemma, and so we conclude that that portion of the diagram commutes. That the lower portion also commutes is easily checked. □

3. push-forwards of O_{F^{r}}-modules

(3.1) The object of this section is to define the push-forward

f_{+} : D_{qc}^{b}(O_{F^{r},Y}) → D_{qc}^{b}(O_{F^{r},X})

for any morphism f : Y → X of smooth k-schemes.

(3.2) As observed in (1.11) and (2.2), the morphism f^{#} : f^{−1}O_{X} → O_{Y} induces a morphism f^{−1}O_{F^{r},X} = f^{−1}O_{X}[F^{r}] → O_{Y} [F^{r}] = O_{F^{r},Y} . Thus if M is any left O_{F^{r},Y} -module, its total push-forward Rf_{∗}M is naturally a complex of O_{F^{r},X}- modules. At first sight this may seem like a reasonable definition of push-forward for O_{F^{r}}-modules. However, it turns out this naive definition is not the appropriate one (except in the case of an étale morphisms f, as we see in lemma 3.5.7 below), and one is led to a more involved construction, which it is the purpose of the rest of this section to describe.

(3.3) We begin by associating an (f^{−1}O_{F^{r},X}, O_{F^{r},Y})-bimodule to any morphism f : Y → X of smooth k-schemes.

Proposition 3.3.1. If f : Y → X is a morphism of smooth k-schemes, so that f^{−1}O_{F^{r},X} ⊗_{f^{−1}O_{X}} ω_{Y/X} is naturally an (f^{−1}O_{F^{r},X}, O_{Y})-bimodule, then the right O_{Y} -module structure extends to a right O_{F^{r},Y} -module structure in such a way that f^{−1}O_{F^{r},X} ⊗_{f^{−1}O_{X}} ω_{Y/X} becomes an (f^{−1}O_{F^{r},X}, O_{F^{r},Y})-bimodule.

Proof. The discussion of the r^{th} relative Frobenius diagram for the morphism f and the r^{th} relative Cartier operator in A.2 shows that the latter is a morphism

C_{Y/X}^{(r)} : F_{Y/X∗}^{(r)}ω_{Y/X} → F_{X}^{r}′^{∗}ω_{Y/X}.

Via the induced bimodule construction of (1.11), we find that f^{−1}O_{F^{r},X} ⊗_{f^{−1}O_{X}} ω_{Y/X} is indeed endowed with the structure of an (f^{−1}O_{F^{r},X}, O_{F^{r},Y})-bimodule. □

Definition 3.3.2. If f : Y → X is a morphism of smooth k-schemes, then the preceding theorem endows f^{−1}O_{F^{r},X} ⊗_{f^{−1}O_{X}} ω_{Y/X} with an (f^{−1}O_{F^{r},X}, O_{F^{r},Y})- bimodule structure. We denote this bimodule by O_{F^{r},X←Y}.

Definition 3.4. Let f : Y → X be a morphism of smooth k-schemes. We define

f_{+} : D_{qc}^{b}(O_{F^{r},Y}) → D_{qc}^{b}(O_{F^{r},X})

as follows: if M^{•} is a complex in D_{qc}^{b}(O_{F^{r},Y}), then

f_{+}M^{•} = Rf_{∗}(O_{F^{r},X←Y} ⊗^{𝕃}_{O_{F^{r} ,Y}} M^{•}).

(3.5) That the functor f_{+} is well-defined will require some justification. We being by proving the following proposition, which will allow us to compute the

derived functor O_{F^{r},X←Y} ⊗^{𝕃}_{O_{F^{r} ,Y}} – occurring in Definition 3.4:

Proposition 3.5.1. If f : Y → X is a morphism of smooth k-schemes, then the (f^{−1}O_{F^{r},X}, O_{F^{r},Y})-bimodule O_{F^{r},X←Y} has a left resolution by (f^{−1}O_{F^{r},X}, O_{F^{r},Y})- bimodules which is bounded and whose terms are flat as right O_{F^{r},Y} -modules.

The proof will follow from a series of lemmas.

Lemma 3.5.2. If f : Y → X is a morphism of smooth k-schemes, then any (f^{−1}O_{F^{r},X}, O_{F^{r},Y})-bimodule which is quasi-coherent as a right O_{F^{r},Y} -module is the quotient of an (f^{−1}O_{F^{r},X}, O_{F^{r},Y})-bimodule which is locally free as a right O_{F^{r},Y} - bimodule

Proof. Let M be an (f^{−1}O_{F^{r},X}, O_{F^{r},Y})-bimodule which is quasi-coherent as a right O_{F^{r},Y} -module. By lemma 1.7.1 (or more properly, its obvious analogue for quasi-coherent right O_{F^{r},Y} -modules), there is a locally free right O_{F^{r},Y} -module P which surjects onto M. Using the left f^{−1}O_{F^{r},X}-module structure of M, we may extend this to a surjection of (f^{−1}O_{F^{r},X}, O_{F^{r},Y})-bimodules

f^{−1}O_{F^{r},X} ⊗_{𝔽_{q}} P → M.

Clearly f^{−1}O_{F^{r},X} ⊗_{𝔽_{q}} P is locally free as a right O_{F^{r},Y} -module, since this is true of P. □

Corollary 3.5.3. If f : Y → X is a morphism of smooth k-schemes, and if U_{i} (i ∈ I) is an open cover of Y such that for each i ∈ I there is a bounded left resolution of the restriction of O_{F^{r},X←Y} to U_{i} by flat right O_{F^{r},U_{i}}-modules, then O_{F^{r},X←Y} has a left resolution by (f^{−1}O_{F^{r},X}, O_{F^{r},Y})-bimodules which is bounded, and whose terms are flat as right O_{F^{r},Y} -modules.

Proof. Since Y is of finite type, we may assume that I is a finite set.

Using the preceding lemma, we may construct a left resolution P^{•} of O_{F^{r},X←Y} by (f^{−1}O_{F^{r},X}, O_{F^{r},Y})-bimodules which are locally free as right O_{F^{r},Y} -modules, which may be of infinite length.

Now standard arguments show that some finite truncation of P^{•} provides a resolution of O_{F^{r},X←Y} by (f^{−1}O_{F^{r},X}, O_{F^{r},Y})-bimodules which are flat as right O_{F^{r},Y} -modules ([Ha, II 4.2]). This proves the corollary. □

Lemma 3.5.4. If f : Y → X and g : Z → Y are morphisms of smooth k-schemes, then there is a natural isomorphism of ((fg)^{−1}O_{F^{r},X}, O_{F^{r},Z})-bimodules,

g^{−1}O_{F^{r},X←Y} ⊗_{g^{−1}O_{F^{r} ,Y}} O_{F^{r},Y ←Z} ⟶^{∼} O_{F^{r},X←Z}.

Proof. We compute

g^{−1}O_{F^{r},X←Y} ⊗_{g^{−1}O_{F^{r} ,Y}} O_{F^{r},Y ←Z}

⟶^{∼} g^{−1}(f^{−1}O_{F^{r},X} ⊗_{f^{−1}O_{X}} ω_{Y/X}) ⊗_{g^{−1}O_{F^{r} ,Y}} g^{−1}O_{F^{r},Y} ⊗_{g^{−1}O_{Y}} ω_{Z/Y}

⟶^{∼} (fg)^{−1}O_{F^{r},X} ⊗_{(fg)^{−1}O_{X}} g^{−1}ω_{Y/X} ⊗_{g^{−1}O_{Y}} ω_{Z/Y}

⟶^{∼} (fg)^{−1}O_{F^{r},X} ⊗_{(fg)^{−1}O_{X}} ω_{Z/X} ⟶^{∼} O_{F^{r},X←Z}.

It is clear that all the natural isomorphisms preserve the left (fg)^{−1}O_{F^{r},X}-module structure. That they also preserve the right O_{F^{r},Z}-module structure follows from the formula for the relative Cartier operator of the composite fg in terms of the relative Cartier operator of the morphisms f and g (A.2.3 (iii)) □

Corollary 3.5.5. If f : Y → X and g : Z → Y are morphisms of smooth kschemes such that O_{F^{r},X←Y} has a bounded left resolution P^{•} by (f^{−1}O_{F^{r},X}, O_{F^{r},Y}) bimodules which are flat as right O_{F^{r},Y} -modules and O_{F^{r},Y ←Z} has a bounded left resolution Q^{•} by (f^{−1}O_{F^{r},Y} , O_{F^{r},Z})-bimodules which are flat as right O_{F^{r},Z}- modules, then O_{F^{r},X←Z} has a left resolution by ((fg)^{−1}O_{F^{r},X}, O_{F^{r},Z})-bimodules which is bounded, and whose terms are flat as right O_{F^{r},Z}-modules.

Proof. The complex g^{−1}P^{•} ⊗_{g^{−1}O_{F^{r} ,Y}} Q^{•} is a complex of ((fg)^{−1}O_{F^{r},X}, O_{F^{r},Z})- bimodules which are flat as right O_{F^{r},Z}-modules. The morphisms P^{•} → O_{F^{r},X←Y} and Q^{•} → O_{F^{r},Y ←Z} induce morphisms

g^{−1}P^{•} ⊗_{g^{−1}O_{F^{r} ,Y}} Q^{•} → g^{−1}P^{•} ⊗_{g^{−1}O_{F^{r} ,Y}} O_{F^{r},Y ←Z}

⟶ g^{−1}O_{F^{r},X←Y} ⊗_{g^{−1}O_{F^{r} ,Y}} O_{F^{r},Y ←Z} ⟶^{∼} O_{F^{r},X←Z}.

The first morphism is a quasi-isomorphism, as g^{−1}P^{•} is a complex of flat g^{−1}O_{F^{r},Y} - modules, and Q^{•} resolves O_{F^{r},Y ←Z}. We must show that the second map is a quasiisomorphism.

Since O_{F^{r},Y ←Z} = g^{−1}O_{F^{r},Y} ⊗_{g^{−1}O_{Y}} ω_{Z/Y} , we may rewrite the second map as

g^{−1}P^{•} ⊗_{g^{−1}O_{Y}} ω_{Z/Y} → g^{−1}O_{F^{r},X←Y} ⊗_{g^{−1}O_{Y}} ω_{Z/Y} .

We now see that this map is a quasi-isomorphism, as g^{−1}P^{•} is a complex of flat g^{−1}O_{Y} -modules and g^{−1}O_{F^{r},X←Y} is also a flat g^{−1}O_{Y} -module. □

Lemma 3.5.6. If X = Spec A is a smooth affine k-scheme, and a is an element of A such that Y = V (a) is also smooth, then O_{F^{r},X←Y} is flat as a right O_{F^{r},Y} - module.

Proof. In the situation of the lemma, the invertible sheaf ω_{Y/X} is freely generated by the section (da)^{−1} over O_{Y} = O_{X}/a. Also O_{Y^{(r)}} = O_{X}/a^{q}, and the Cartier operator is the morphism b(da)^{−1} ↦ F^{r}ba^{q−1}(da)^{−1} (A.2.3 (iv)), where the right hand side is viewed as element of f^{−1}O_{F^{r},X} ⊗ ω_{Y/X}. Thus

O_{F^{r},X←Y} = f^{−1}O_{F^{r},X} ⊗_{f^{−1}O_{X}} ω_{Y/X} ⟶^{∼} f^{−1}O_{X}[F^{r}] ⊗_{f^{−1}O_{X}} (O_{X}/a)(da)^{−1}

⟶^{∼} ⨁_{n=0}^{∞} O_{X}/a^{q^{n}}F^{rn} (da)^{−1},

with the right O_{F^{r},Y} -module structure on O_{F^{r},X←Y} defined by

bF^{rn} (da)^{−1} ↦^{·F^{r}} ba^{q^{n}(q−1)}F^{rn+r} (da)^{−1}.

For each i ≥ 0, set

M_{i} = ⨁_{n=0}^{∞} a^{max(0,q^{n}−i)}O_{X}/a^{q^{n}}F^{rn} (da)^{−1}.

Then each M_{i} is a right O_{F^{r},Y} -submodule of O_{F^{r},X←Y} , and O_{F^{r},X←Y} = lim_{→}_{i}M_{i}. We claim that each M_{i} is free as a right O_{F^{r},Y} -module. Once we prove this, the lemma will follow, since a direct limit of free right O_{F^{r},Y} -modules is a flat right O_{F^{r},Y} -module.

It will suffice to show that each quotient M_{i}/M_{i+1} is free. Let m be the least integer greater than or equal to log_{q} i. Then

M_{i}/M_{i+1} = ⨁_{n=0}^{∞} a^{max(0,q^{n}−i)}O_{X}/a^{max(0,q^{n}−i+1)}F^{rn} (da)^{−1}

= ⨁_{n=m}^{∞} a^{q^{n}−i}O_{X}/a^{q^{n}−i+1}F^{rn} (da)^{−1},

and this is a free right O_{F^{r},Y} = (O_{X}/a)[F^{r}]-module of rank one, freely generated by F^{m}. This completes the proof of the lemma. □

Lemma 3.5.7. If f : Y → X is an étale morphism of smooth k-schemes, then O_{F^{r},X←Y} is isomorphic to O_{F^{r},Y} , regarded as a right O_{F^{r},Y} -module in the obvious way, and as a left f^{−1}O_{F^{r},X}-module via the natural morphism f^{−1}O_{F^{r},X} → O_{F^{r},Y} . In particular O_{F^{r},X←Y} is flat as a right O_{F^{r},Y} -module.

Proof. In the situation of the lemma we see that ω_{Y/X} ⟶^{∼} O_{Y} , that the morphism F_{Y/X}^{(r)} is the identity morphism of Y, and that F_{X}^{r}′ = F_{Y}^{r} is the Frobenius morphism of Y. There are thus canonical isomorphisms

F_{Y/X∗}^{(r)}ω_{Y/X} = ω_{Y/X} = O_{Y} and F_{X}^{r}′^{∗}ω_{Y/X} = F_{Y}^{r} ^{∗}O_{Y} = O_{Y} ,

and with respect to these isomorphisms the Cartier operator is simply the identity morphism of O_{Y} . The lemma follows immediately from these remarks. □

Corollary 3.5.8. If f : Y → X is an immersion, then O_{F^{r},X←Y} is flat as a right O_{F^{r},Y} -module.

Proof. Lemma 3.5.4 shows that if f_{1} and f_{2} are two immersions for which the corollary holds true, and the source of f_{1} equals the target of f_{2}, then the corollary also holds true for the composite immersion f_{1}f_{2}.

In particular, since we may factor f as the composition of an open immersion and a closed immersion, it suffices to prove the corollary for each kind of immersion separately.

Since open immersions are étale, the corollary follows in this case by lemma 3.5.7, and so it remains to consider closed immersions. The property of flatness can be verified locally, and locally we may factor a closed immersion into a composition of closed immersions of the kind considered in lemma 3.5.6 above. Thus that lemma proves the corollary in the case of closed immersions. □

Lemma 3.5.9. Let X = Spec A be a smooth affine k-scheme such that O_{X}^{(r)} is free as a right O_{X}-module, and let f : Y = X ×_{k}𝔾_{m} → X be the projection onto the first factor. Then O_{F^{r},X←Y} has a bounded left resolution by free right O_{F^{r},Y} -modules.

Proof. Rather than considering 𝔾_{m} as a scheme over k, and taking the cartesian product with X over k, it will be more convenient to regard 𝔾_{m} as a scheme over 𝔽_{p} and to take the cartesian product over 𝔽_{p}. Thus write 𝔾_{m} = Spec 𝔽_{p} [t, t^{−1}]. Let g : Y = X ×_{𝔽_{p}} 𝔾_{m} → 𝔾_{m} denote the projection onto the second factor. Then O_{Y} = f^{−1}O_{X} ⊗_{𝔽_{p}} g^{−1}O_{𝔾_{m}}, and

ω_{Y/X} ⟶^{∼} O_{Y} (dt/t) ⟶^{∼} f^{−1}O_{X} ⊗_{𝔽_{p}} g^{−1}O_{𝔾_{m}} (dt/t).

If b is a section of O_{𝔾_{m}}, then we may write b = ∑_{i=0}^{q−1} t^{i}b′_{i}^{q} for some uniquely determined sections b′_{i} of O_{𝔾_{m}}. The relative Cartier operator F_{Y/X∗}^{(r)}ω_{Y/X} → F_{X}^{r}′^{∗}ω_{Y/X} is determined by the formula

a ⊗ b dt/t ↦ a ⊗ b′_{0}dt/t

(A.2.3 (v)), where a is any section of O_{X}. From this we see that the right O_{F^{r},Y} - module structure of

O_{F^{r},X←Y} ⟶^{∼} f^{−1}O_{F^{r},X} ⊗_{f^{−1}O_{X}} ω_{Y/X} ⟶^{∼} f^{−1}O_{F^{r},X} ⊗_{k} g^{−1}O_{𝔾_{m}} dt/t

is determined by the multiplication rule

1 ⊗ b dt/t ↦^{·F^{r}} F^{r} ⊗ b′_{0}dt/t.

From this formula we see that the morphism

(3.5.10)  O_{F^{r},Y} → O_{F^{r},X←Y}

defined by P ↦ (1 ⊗ dt/t) · P is a surjection of right O_{F^{r},Y} -modules. We will show that its kernel is a free O_{F^{r},Y} -module, thereby proving the lemma.

Since O_{X}^{(1)} is free as a right O_{X}-module, the same is true of O_{X}^{(n)} for each positive integer n. Let {a_{n,j}|j ∈ J_{n}} be a basis for O_{X}^{(rn)} as a free right O_{X}-module.

Now we see that

O_{F^{r},𝔾_{m}} = O_{𝔾_{m}}[F^{r}] = ⨁_{n=0}^{∞} F^{rn}O_{𝔾_{m}} ⊕ ⨁_{n=1}^{∞} ⨁_{i=1}^{q−1} F^{rn−r}t^{i}F^{r}O_{𝔾_{m}}[F^{r}]

In the case when k = 𝔽_{p} and X is a point, the kernel of (3.5.10) is precisely the second summand above. In general tensoring the above equation by f^{−1}O_{X} over 𝔽_{p} and using the isomorphism O_{F^{r},Y} ⟶^{∼} f^{−1}O_{X} ⊗_{𝔽_{p}} g^{−1}O_{𝔾_{m}}[F^{r}] we see that the required kernel is equal to

⨁_{n=1}^{∞} ⨁_{i=1}^{q−1} ⨁_{j∈J_{n}} a_{n,j}F^{rn−r}t^{i}F^{r}O_{F^{r},Y}

which is a free right O_{F^{r},Y} -submodule of O_{F^{r},Y} . □

Corollary 3.5.11. If f : Y → X is a smooth morphism of smooth k-schemes, then O_{F^{r},X←Y} has a bounded left resolution by (f^{−1}O_{F^{r},X}, O_{F^{r},Y})-bimodules which are flat as right O_{F^{r},Y} -modules.

Proof. Corollary 3.5.3 shows that it suffices to prove the result locally on Y, and corollary 3.5.5 shows that if we factor f as a composition of morphisms, it suffices to prove the result for each of these morphisms separately.

Locally, any smooth morphism may be factored as a composition of an étale morphism and a sequence of projections U ×_{k} 𝔾_{m} → U. Lemmas 3.5.7 and 3.5.9 establish the corollary for each of these kinds of morphisms. (In order to apply 3.5.9 we may have to localize further, so that the locally free sheaf O_{X}^{(1)} becomes free.) □

(3.5.12) Proof of 3.5.1 The morphism f : Y → X factors as

Y ⟶^{Γ_{f}} X ×_{k} Y ⟶^{p_{1}} X,

where Γ_{f} is the graph of f and p_{1} is the projection onto the first factor. That the proposition holds for the morphisms Γ_{f} and p_{1} follows from corollaries 3.5.8 and 3.5.11 respectively. Corollary 3.5.5 now shows that the proposition holds for f = p_{1}Γ_{f}. □

Corollary 3.5.13. Let f : Y → X be a morphism of smooth k-schemes. If M^{•} is an object of D_{qc}^{b}(O_{F^{r},Y}) then M^{•} is quasi-isomorphic to a bounded complex of modules which are acyclic for the functor O_{F^{r},X←Y} ⊗_{O_{F^{r} ,Y}} (–).

Proof. Lemma 1.7.1 provides a resolution bounded above of M^{•} by locally free O_{F^{r},Y} -modules, which are certainly acyclic for O_{F^{r},X←Y} ⊗_{O_{F^{r} ,Y}} (–). Proposition 3.5.1 shows that O_{F^{r},X←Y} ⊗_{O_{F^{r} ,Y}} (–) has finite cohomological dimension. Thus some (sufficiently long) finite truncation of this locally free resolution will also consist of modules acyclic for O_{F^{r},X←Y} ⊗_{O_{F^{r} ,Y}} (–). □

(3.5.14) We apply the above to study our proposed definition of f_{+}. First note that by lemma 3.5.2 there is a well defined functor.

O_{F^{r},X←Y} ⊗^{𝕃}_{O_{F^{r} ,Y}} – : D^{−}(O_{F^{r},Y}) → D^{−}(f^{−1}O_{F^{r},X}).

By proposition 3.5.1, it restricts to

O_{F^{r},X←Y} ⊗^{𝕃}_{O_{F^{r} ,Y}} – : D^{b}(O_{F^{r},Y}) → D^{b}(f^{−1}O_{F^{r},X}).

Since Rf_{∗} has cohomological amplitude at most d_{Y} ([Ha2, III 2.7]) we obtain a functor Rf_{∗} : D^{b}(f^{−1}O_{F^{r},X}) → D^{b}(O_{F^{r},Y}). Composing these two derived functors yields the functor

Rf_{∗}(O_{F^{r},X←Y} ⊗^{𝕃}_{O_{F^{r} ,Y}} –) : D^{b}(O_{F^{r},Y}) ⟶ D^{b}(O_{F^{r},X}).

Note that if M^{•} is any complex of O_{F^{r},Y} modules, it is crucial when computing

f_{+} that we apply f_{∗} to a resolution of O_{F^{r},X←Y} ⊗^{𝕃}_{O_{F^{r} ,Y}} M^{•} by flasque f^{−1}O_{F^{r},X}- modules. One cannot apply f_{∗} to a flaque resolution by f^{−1}O_{X}-modules and then appeal to functoriality to obtain an action of O_{F^{r},X}, since this procedure would only produce an object of D_{qc}^{b}(O_{X}) equipped with an O_{F^{r},X}-action (so to speak), which does not specify a unique object of D_{qc}^{b}(O_{F^{r},X}).

Thus to show that f_{+} is well defined, it remains to show that it take complexes with O_{F^{r},Y} - quasi-coherent cohomology to complexes with O_{F^{r},X} quasi-coherent cohomology. Remark 3.5.15 Note that if f : Y → X is an immersion, then by corollary 3.5.8, O_{F^{r},X←Y} is flat as a right O_{F^{r},Y} -module. Thus in this case, there is no need to consider derived functors, and we may define the functor

O_{F^{r},X←Y} ⊗_{O_{F^{r} ,Y}} – : D^{•}(O_{F^{r},Y}) → D^{•}(f^{−1}O_{F^{r},X}),

where • is any one of +, −, b or ∅.

(3.6) To show that f_{+} preserves the property of having quasi-coherent cohomology we need two simple lemmas.

Lemma 3.6.1. If f : Y → X is a morphism of smooth k-schemes, then any induced left O_{F^{r},Y} -module is acyclic for the right-exact functor O_{F^{r},X←Y} ⊗_{O_{F^{r} ,Y}} –.

Proof. Let M be an O_{Y} -module, and M = O_{F^{r},Y} ⊗_{O_{Y}} M the corresponding induced left O_{F^{r},Y} -module. By lemma 1.3.1, O_{F^{r},X} is locally free as a right O_{X}-module. Thus f^{−1}O_{F^{r},X} is locally free as a right f^{−1}O_{X}-module. Since ω_{Y/X} is an invertible sheaf on Y, we conclude that

O_{F^{r},X←Y} = f^{−1}O_{F^{r},X} ⊗_{f^{−1}O_{X}} ω_{Y/X}

is locally free as an O_{Y} -module. We now compute

O_{F^{r},X←Y} ⊗^{𝕃}_{O_{F^{r} ,Y}} M ⟶^{∼} O_{F^{r},X←Y} ⊗^{𝕃}_{O_{F^{r} ,Y}} O_{F^{r},Y} ⊗_{O_{Y}} M

⟶^{∼} O_{F^{r},X←Y} ⊗^{𝕃}_{O_{Y}} M ⟶^{∼} O_{F^{r},X←Y} ⊗_{O_{Y}} M,

proving that M is acyclic for O_{F^{r},X←Y} ⊗_{O_{F^{r} ,Y}} –. □

Lemma 3.6.2. If f : Y → X is a morphism of smooth k-schemes and M is any O_{Y} -module, then the natural map

O_{F^{r},X} ⊗_{O_{X}} Rf_{∗}M → Rf_{∗}(f^{−1}O_{F^{r},X} ⊗_{f^{−1}O_{X}} M)

is an isomorphism.

Proof. By lemma 1.3.1, O_{F^{r},X} = ⨁_{n=0}^{∞} F_{X∗}^{rn}O_{X} is a locally free right O_{X}-module. The lemma is thus a special case of the projection formula. □

We are now ready to prove that f_{+} preserves the property of having quasicoherent cohomology modules.

Theorem 3.6.3. Let f : Y → X be a morphism of smooth k-schemes. Then the functor

f_{+}(–) = Rf_{∗}(O_{F^{r},X←Y} ⊗^{𝕃}_{O_{F^{r} ,Y}} –) : D^{b}(O_{F^{r},Y}) → D^{b}(O_{F^{r},Y})

restricts to a functor

f_{+}(–) = Rf_{∗}(O_{F^{r},X←Y} ⊗^{𝕃}_{O_{F^{r} ,Y}} –) : D_{qc}^{b}(O_{F^{r},Y}) → D_{qc}^{b}(O_{F^{r},Y}).

Proof. Let M^{•} be a complex in D_{qc}^{b}(O_{F^{r},Y}). Then by lemma 1.10.1, we may find a left resolution N^{•} of M^{•} by induced left O_{F^{r},Y} -modules

N^{•} = O_{F^{r},Y} ⊗_{O_{Y}} N^{•},

for some quasi-coherent O_{Y} -module N^{•}, which is bounded above. By lemma 3.6.1 these modules are acyclic for the functor O_{F^{r},X←Y} ⊗_{O_{F^{r} ,Y}} (–). Thus

O_{F^{r},X←Y} ⊗^{𝕃}_{O_{F^{r} ,Y}} M^{•} = O_{F^{r},X←Y} ⊗_{O_{F^{r} ,Y}} N^{•}

is a complex of left f^{−1}O_{F^{r},X}-modules, which is bounded above.

Consider a double complex C^{•,•} of Rf_{∗}-acyclic f^{−1}O_{F^{r},X}-modules which resolves O_{F^{r},X←Y} ⊗_{O_{F^{r} ,Y}} N^{•} (which may be constructed, for example, by using Godement resolutions). Then we have

f_{+}M = Rf_{∗}(O_{F^{r},X←Y} ⊗_{O_{F^{r} ,Y}} N^{•}) = f_{∗}C^{T ot,•},

where C^{T ot,•} denotes the total complex of C^{•,•}. (This is not completely trivial, because C^{T ot,•} may be unbounded below, and we are applying the right derived functor Rf_{∗} to it. However, it follows from the fact that C^{T ot,•} is quasi-isomorphic to the complex M^{•}, together with the fact that Rf_{∗} has finite cohomological dimension, so that it makes sense to apply it to complexes bounded above.) For any fixed value of i we see that

f_{∗}C^{i,•} = Rf_{∗}(O_{F^{r},X←Y} ⊗_{O_{F^{r} ,Y}} N^{i}) = f_{+}N^{i}.

Thus we obtain a spectral sequence

E_{1}^{i,j} = H^{j}(f_{+})N^{i} ⇒ H^{i+j}(f_{+})M^{•}.

Since Rf_{∗} has finite cohomological amplitude, this spectral sequence converges. Hence in order to show that the cohomology sheaves of f_{+}M^{•} are quasi-coherent, it suffices to show that the cohomology sheaves of f_{+}N^{i} are quasi-coherent.

Now lemma 3.6.2 shows that

H^{j}(f_{+})N^{i} = Rf_{∗}^{j}(O_{F^{r},X←Y} ⊗_{O_{F^{r} ,Y}} N^{i})

= Rf_{∗}^{j}(f^{−1}O_{F^{r},X} ⊗_{f^{−1}O_{X}} ω_{Y/X} ⊗_{O_{F^{r} ,Y}} O_{F^{r},Y} ⊗_{O_{Y}} N^{i})

= Rf_{∗}^{j}(f^{−1}O_{F^{r},X} ⊗_{f^{−1}O_{X}} ω_{Y/X} ⊗_{O_{Y}} N^{i})

⟶^{∼} O_{F^{r},X} ⊗_{O_{X}} Rf_{∗}^{j}(ω_{Y/X} ⊗_{O_{Y}} N^{i}).

Since ω_{Y/X} ⊗_{O_{Y}} N^{i} is a quasi-coherent O_{Y} -module, we see that Rf_{∗}^{j}(ω_{Y/X} ⊗_{O_{Y}} N^{i}) is a quasi-coherent O_{X}-module. Thus f_{+}N^{i} does indeed have quasi-coherent cohomology sheaves, for any i, and so the same is true of f_{+}M^{•}. □

(3.7) The proof of theorem 3.6.3 shows that induced modules play an important role in computing push-forwards. For this reason it will be useful to answer the following question: suppose that β′ : O_{F^{r},Y} ⊗_{O_{Y}} M → O_{F^{r},Y} ⊗_{O_{Y}} N is a morphism of induced left O_{F^{r},Y} -modules. How then can one describe the induced morphism

O_{F^{r},X} ⊗_{O_{X}} Rf_{∗}(ω_{Y/X} ⊗_{O_{Y}} M) ⟶^{∼} f_{+}(O_{F^{r},Y} ⊗_{O_{Y}} M)

⟶^{f_{+}β′} f_{+}(O_{F^{r},Y} ⊗_{O_{Y}} N) ⟶^{∼} O_{F^{r},X} ⊗_{O_{X}} Rf_{∗}(ω_{Y/X} ⊗_{O_{Y}} N) ?

As observed in 1.10.3, to give the morphism β′ is equivalent to giving a morphism

β : M → ⨁_{n=0}^{∞}(F_{Y}^{rn})^{∗}N

of O_{Y} -modules. Since f_{+} commutes with direct sums of morphisms, we may deal with each summand individually. Our question is then answered by the following result:

Proposition 3.7.1. Let f : Y → X be a morphism of smooth k-schemes. Suppose given O_{Y} -modules M and N and a morphism

β : M → (F_{X}^{rn})^{∗}N

for some n, and let β′ : O_{F^{r},Y} ⊗_{O_{Y}} M → O_{F^{r},Y} ⊗_{O_{Y}} N be the corresponding morphism of induced left O_{F^{r},Y} -modules (via 1.10.3).

Then β gives rise to a natural morphism

γ : Rf_{∗}(ω_{Y/X} ⊗_{O_{Y}} M) → (F_{X}^{rn})^{∗}Rf_{∗}(ω_{Y/X} ⊗_{O_{Y}} N),

such that, if

γ′ : O_{F^{r},X} ⊗_{O_{X}} Rf_{∗}(ω_{Y/X} ⊗_{O_{Y}} M) → O_{F^{r},X} ⊗_{O_{X}} Rf_{∗}(ω_{Y/X} ⊗_{O_{Y}} N)

denotes the corresponding morphism of induced left O_{F^{r},X}-modules, then under the natural identifications

f_{+}(O_{Y,F} ⊗_{O_{Y}} M) ⟶^{∼} O_{F^{r},X} ⊗_{O_{X}} Rf_{∗}(ω_{Y/X} ⊗_{O_{Y}} M)

and

f_{+}(O_{Y,F} ⊗_{O_{Y}} N) ⟶^{∼} O_{F^{r},X} ⊗_{O_{X}} Rf_{∗}(ω_{Y/X} ⊗_{O_{Y}} N)

(inverse to the isomorphisms provided by lemma 3.6.2), the morphism γ′ coincides with the morphism f_{+}β′.

Proof. We will construct the morphism γ, but leave it to the reader to check the asserted equality γ′ = f_{+}β′, since it consists simply of chasing through the definitions. We use the notation and terminology of the rn^{th} Frobenius diagram of X. (See (A.2)). In particular, the rn^{th} relative Cartier operator is an O_{Y^{(rn)}}-linear morphism F_{Y/X∗}^{(rn)}ω_{Y/X} → (F_{X}^{rn})′^{∗}ω_{Y/X}. Combining this with the morphism β and the

projection formula for the affine morphism F_{Y/X}^{(rn)} we obtain the following sequence of morphisms:

F_{Y/X∗}^{(rn)}(ω_{Y/X} ⊗_{O_{Y}} M) ⟶ F_{Y/X∗}^{(rn)}(ω_{Y/X} ⊗_{O_{Y}} (F_{Y}^{rn})^{∗}N)

⟶^{∼} F_{Y/X∗}^{(rn)}(ω_{Y/X} ⊗_{O_{Y}} (F_{Y/X}^{(rn)})^{∗}(F_{X}^{rn})′^{∗}N) ⟶^{∼} F_{Y/X∗}^{(rn)}ω_{Y/X} ⊗_{O_{Y ^{(rn)}}} (F_{X}^{rn})′^{∗}N

⟶ (F_{X}^{rn})′^{∗}ω_{Y/X} ⊗_{O_{Y ^{(rn)}}} (F_{X}^{rn})′^{∗}N ⟶^{∼} (F_{X}^{rn})′^{∗}(ω_{Y/X} ⊗_{O_{Y}} N).

Applying Rf_{∗}^{(rn)} to the source and target of the composite of this sequence of morphisms, and applying base-change for the flat map F_{X}^{rn}, yields the required morphism γ:

Rf_{∗}(ω_{Y/X} ⊗_{O_{Y}} M) ⟶^{∼} Rf_{∗}^{(rn)}F_{Y/X∗}^{(rn)}(ω_{Y/X} ⊗_{O_{Y}} M)

⟶ Rf_{∗}^{(rn)}(F_{X}^{rn})′^{∗}(ω_{Y/X} ⊗_{O_{Y}} N) ⟶^{∼} (F_{X}^{rn})^{∗}Rf_{∗}(ω_{Y/X} ⊗_{O_{Y}} N).

This is the required morphism γ. □

Proposition 3.8. Suppose that g : Z → Y and f : Y → X are two morphisms of smooth k-schemes. Then there is a natural isomorphism

f_{+}g_{+} ⟶^{∼} (fg)_{+}.

Proof. Lemma 3.5.4 provides an isomorphism

O_{F^{r},X←Z} ⟶^{∼} g^{−1}O_{F^{r},X←Y} ⊗_{g^{−1}O_{F^{r} ,Y}} O_{F^{r},Y ←Z}.

Just as in the proof of corollary 3.5.5, let P^{•} and Q^{•} be the finite length right-flat resolutions of O_{F^{r},X←Y} and O_{F^{r},Y ←Z} whose existence is asserted by proposition 3.5.1. Then for any object M^{•} of D^{b}(O_{F^{r},Z})

(fg)_{+}M^{•} = R(fg)_{∗}(O_{F^{r},X←Z} ⊗^{𝕃}_{O_{F^{r} ,Z}} M^{•})

⟶^{∼} Rf_{∗}Rg_{∗}(g^{−1}P^{•} ⊗_{g^{−1}O_{F^{r} ,Y}} Q^{•} ⊗_{O_{F^{r} ,Z}} M^{•})

⟶^{∼} Rf_{∗}(P^{•} ⊗_{O_{F^{r} ,Y}} Rg_{∗}(Q^{•} ⊗_{O_{F^{r} ,Z}} M^{•}))

= Rf_{∗}(O_{F^{r},X←Y} ⊗^{𝕃}_{O_{F^{r} ,Y}} Rg_{∗}(O_{F^{r},Y ←Z} ⊗^{𝕃}_{O_{F^{r} ,Z}} M^{•}))

= f_{+}g_{+}M^{•}.

(Here we have applied the projection formula for the complex P^{•} and the morphism g. Although the complex P^{•} is not necessarily locally free, it is quasi-isomorphic to a bounded above locally free complex. Since Rg_{∗} has finite cohomological dimension, we may apply it to bounded above complexes, and so replacing P^{•} by such a locally free resolution, we see that the projection formula is valid for the complex P^{•}.) Thus the proposition is proved. □

Proposition 3.9. Let f : Y → X be a morphism of smooth k-schemes and g : U → X be an open immersion. Form the cartesian diagram

\xymatrix{V = g^{−1}(U) \ar[r]^{g′} \ar[d]^{f′} & Y \ar[d]^{f} \\ U \ar[r]^{g} & X.}

Then for any object M^{•} of D_{qc}^{b}(O_{F^{r},Y}) there is a natural isomorphism

g^{!}f_{+}M^{•} ⟶^{∼} f′_{+}g^{′!}M^{•}.

Proof. This follows from the fact that pull-back by an open immersion consists simply of restricting to the source of the immersion, that the formation of Rf_{∗} is local on the base, and that the construction of O_{F^{r},X←Y} is local on the target. □

Proposition 3.10. Let f : Y → X be a morphism of smooth k-schemes, let k′/k be a field extension, and let f′ : Y′ → X′ be the base-change of f over k. Then the diagram

\xymatrix{D_{qc}^{b}(O_{F^{r},Y}) \ar[d]^{f_{+}} \ar[r]^{k′⊗_{k}–} & D_{qc}^{b}(O_{F^{r},Y′}) \ar[d]^{f′_{+}} \\ D_{qc}^{b}(O_{F^{r},X}) \ar[r]^{k′⊗_{k}–} & D_{qc}^{b}(O_{F^{r},X′})}

commutes up to natural isomorphism, in a manner compatible with the natural isomorphisms of propositions 3.8 and 3.9.

Proof. This follows from the standard compatibilities satisfied by tensor products, together with flat base-change for the total derived push-forward of quasi-coherent modules (applied to the base-change from k′ to k) and the natural isomorphism

k′ ⊗_{k} ω_{Y/X} ⟶^{∼} ω_{Y′/X′}. □

4. relations between f_{+} and f^{!}

(4.1) In this section we prove a projection formula relating f_{+} and f^{!} for any morphism f : Y → X of smooth k-schemes, as well as adjointness formulae relating f_{+} and f^{!} in the cases when f is proper or an open immersion.

We begin with the projection formula, which is straightforward to prove:

Proposition 4.2. If f : Y → X is a morphism of smooth k-schemes, let M^{•} be an object of D_{qc}^{b}(O_{F^{r},Y}) and N^{•} be an object of D_{qc}^{b}(O_{F^{r},X}). Then there is a natural isomorphism

f_{+}(M^{•} ⊗^{𝕃}_{O_{Y}} f^{!}N^{•}) ⟶^{∼} f_{+}M^{•}[d_{Y/X}] ⊗^{𝕃}_{O_{X}} N^{•}.

Proof. We may assume that N^{•} is a bounded complex of left O_{F^{r},X}-modules which are flat as O_{X}-modules. Then f^{!}N^{•} is represented by

O_{Y} ⊗_{f^{−1}O_{X}} f^{−1}N^{•}[d_{Y/X}].

Observe also that this is a bounded complex of left O_{F^{r},Y} -modules which are flat as O_{Y} -modules. Thus

f_{+}(M^{•} ⊗^{𝕃}_{O_{Y}} f^{!}N^{•}) ⟶^{∼} Rf_{∗}(O_{F^{r},X←Y} ⊗^{𝕃}_{O_{F^{r} ,Y}} M^{•}⊗_{O_{Y}} O_{Y} ⊗_{f^{−1}O_{X}} f^{−1}N^{•}[d_{Y/X}])

⟶^{∼} Rf_{∗}(O_{F^{r},X←Y} ⊗^{𝕃}_{O_{F^{r} ,Y}} M^{•} ⊗_{f^{−1}O_{X}} f^{−1}N^{•}[d_{Y/X}])

⟶^{∼}^{(1)} Rf_{∗}(O_{F^{r},X←Y} ⊗^{𝕃}_{O_{F^{r} ,Y}} M^{•}[d_{Y/X}]) ⊗_{O_{X}} N^{•}

⟶^{∼} f_{+}M^{•}[d_{Y/X}] ⊗^{𝕃}_{O_{X}} N^{•}.

(Isomorphism (1) follows from the projection formula applied to the complex of locally free O_{X}-modules N^{•}.) This proves the proposition. □

(4.3) We now prove the adjointness between f_{+} and f^{!} in the rather simple case of an open immersion. In fact it will follow immediately from the fact that in this case one can describe f_{+} and f^{!} in terms of the usual pull-back and push-forward of sheaves.

Lemma 4.3.1. If f : Y → X is an open immersion of smooth k-schemes, then the functor

f^{!} : D_{qc}^{b}(O_{F^{r},X}) → D_{qc}^{b}(O_{F^{r},Y})

is naturally isomorphic to the functor f^{−1}, while the functor

f_{+} : D_{qc}^{b}(O_{F^{r},Y}) → D_{qc}^{b}(O_{F^{r},X})

is naturally isomorphic to the functor Rf_{∗}. Thus if M^{•} is any object of D_{qc}^{b}(O_{F^{r},Y}) and N^{•} any object of D_{qc}^{b}(O_{F^{r},Y}), there is a natural isomorphism of objects in D^{+}(X):

\underline{RHom}_{O_{F^{r} ,X}}^{•} (f_{+}M^{•}, N^{•}) ⟶^{∼} Rf_{∗}\underline{RHom}_{O_{F^{r} ,Y}}^{•} (M^{•}, f^{!}N^{•}).

In particular (take degree zero cohomology of the right derived functor of global sections) the functor f_{+} is right adjoint to the functor f^{!}, and the resulting natural transformation

f^{!}f_{+} → id

is an isomorphism of functors.

Proof. The descriptions of f_{+} and f^{!} follow directly from the definitions, since when f is an open immersion, f^{−1}O_{F^{r},X} = O_{F^{r},Y} and ω_{Y/X} = f^{−1}O_{X} = O_{Y} . The consequent adjointness is a standard property of the functors Rf_{∗} and f^{−1}. □

(4.4) We now turn to the theorem which expresses the adjointness of the morphisms f_{+} and f^{!} in the sense of derived categories, for the case of a proper morphism:

Theorem 4.4.1. (i) Let f : Y → X be a proper morphism of smooth k-schemes. If M^{•} is any object of D_{qc}^{b}(O_{F^{r},Y}) and N^{•} any object of D_{qc}^{b}(O_{F^{r},Y}), there is a natural isomorphism of objects in D^{+}(X):

\underline{RHom}_{O_{F^{r} ,X}}^{•} (f_{+}M^{•}, N^{•}) ⟶^{∼} Rf_{∗}\underline{RHom}_{O_{F^{r} ,Y}}^{•} (M^{•}, f^{!}N^{•}).

In particular (take degree zero cohomology of the right derived functor of global sections) the functor f_{+} is left adjoint to the functor f^{!}.

(ii) Let f : Y → X be a proper morphism of smooth k-schemes and g : U → X be an open immersion. Form the cartesian diagram

\xymatrix{V = g^{−1}(U) \ar[r]^{g′} \ar[d]^{f′} & Y \ar[d]^{f} \\ U \ar[r]^{g} & X.}

Then for any objects M^{•} of D_{qc}^{b}(O_{F^{r},Y}) and N^{•} of D_{qc}^{b}(O_{F^{r},X}) the diagram of natural isomorphisms

\xymatrix{g^{−1}\underline{RHom}_{O_{F^{r} ,X}}^{•} (f_{+}M^{•}, N^{•}) \ar[r]^{∼}_{part (i)} \ar[d]^{∼} & g^{−1}Rf_{∗}\underline{RHom}_{O_{F^{r} ,Y}}^{•} (M^{•}, f^{!}N^{•}) \ar[d]^{∼} \\ \underline{RHom}_{O_{F^{r} ,U}}^{•} (g^{−1}f_{+}M^{•}, g^{−1}N^{•}) \ar[d]^{∼}_{(4.3.1)} & Rf′_{∗}g^{′−1}\underline{RHom}_{O_{F^{r} ,Y}}^{•} (M^{•}, f^{!}N^{•}) \ar[d]^{∼} \\ \underline{RHom}_{O_{F^{r} ,U}}^{•} (g^{!}f_{+}M^{•}, g^{!}N^{•}) \ar[d]^{∼}_{(3.9)} & Rf′_{∗}\underline{RHom}_{O_{F^{r} ,V}}^{•} (g^{−1}M^{•}, g^{−1}f^{!}N^{•}) \ar[d]^{∼}_{(4.3.1)} \\ \underline{RHom}_{O_{F^{r} ,U}}^{•} (f′_{+}g^{′!}M^{•}, g^{!}N^{•}) \ar@{-->}[dr]^{∼}_{part (i)} & Rf′_{∗}\underline{RHom}_{O_{F^{r} ,V}}^{•} (g^{′!}M^{•}, g^{′!}f^{!}N^{•}) \ar[d]^{∼}_{(2.4)} \\  & Rf′_{∗}\underline{RHom}_{O_{F^{r} ,V}}^{•} (g^{′!}M^{•}, f^{′!}g^{!}N^{•})}

(in which the isomorphisms are labelled by the result which gives rise to them) commutes. In other words, the adjointness of (i) is local on the base.

(iii) Let f : Y → X and g : Z → Y be proper morphisms of smooth k-schemes. Then for any objects M^{•} of D_{qc}^{b}(O_{F^{r},Z}) and N^{•} of D_{qc}^{b}(O_{F^{r},X}) the diagram of natural isomorphisms

\xymatrix{\underline{RHom}_{O_{F^{r} ,X}}^{•} ((fg)_{+}M^{•}, N^{•}) \ar[d]^{∼}_{(3.8)} \ar[r]^{∼}_{part (i)} & R(fg)_{∗}\underline{RHom}_{O_{F^{r} ,Z}}^{•} (M^{•}, (fg)^{!}N^{•}) \ar[dd]^{∼}_{(2.4)} \\ \underline{RHom}_{O_{F^{r} ,X}}^{•} (f_{+}g_{+}M^{•}, N^{•}) \ar[d]^{∼}_{part (i)} \\ Rf_{∗}\underline{RHom}_{O_{F^{r} ,Y}}^{•} (g_{+}M^{•}, f^{!}N^{•}) \ar[r]^{∼}_{part (i)} & Rf_{∗}Rg_{∗}\underline{RHom}_{O_{F^{r} ,Z}}^{•} (M^{•}, g^{!}f^{!}N^{•})}

(in which the isomorphisms are labelled by the results which give rise to them) commutes. In other words, the adjointness of (i) is compatible with compositions.

We begin by proving the necessary preliminary results.

Proposition 4.4.2. (i) Let f : Y → X be a morphism of smooth k-schemes, and let M^{•} and N^{•} be objects of D_{qc}^{b}(O_{F^{r},Y}). Then there is a natural transformation

Rf_{∗}\underline{RHom}_{O_{F^{r} ,Y}}^{•} (M^{•}, N^{•}) → \underline{RHom}_{O_{F^{r} ,X}}^{•} (f_{+}M^{•}, f_{+}N^{•}).

(ii) Let f : Y → X be a morphism of smooth k-schemes and g : U → X be an open immersion. Form the cartesian diagram

\xymatrix{V = g^{−1}(U) \ar[r]^{g′} \ar[d]^{f′} & Y \ar[d]^{f} \\ U \ar[r]^{g} & X.}

Then for any objects M^{•} and N^{•} of D_{qc}^{b}(O_{F^{r},Y}) the diagram of natural isomorphisms

\xymatrix{g^{−1}Rf_{∗}\underline{RHom}_{O_{F^{r} ,Y}}^{•} (M^{•}, N^{•}) \ar[r]_{part (i)} \ar[d]^{∼} & g^{−1}\underline{RHom}_{O_{F^{r} ,X}}^{•} (f_{+}M^{•}, f_{+}N^{•}) \ar[d]^{∼} \\ Rf′_{∗}g^{′−1}\underline{RHom}_{O_{F^{r} ,Y}}^{•} (M^{•}, N^{•}) \ar[d]^{∼} & \underline{RHom}_{O_{F^{r} ,U}}^{•} (g^{−1}f_{+}M^{•}, g^{−1}f_{+}N^{•}) \ar[d]^{∼}_{(4.3.1)} \\ Rf′_{∗}\underline{RHom}_{O_{F^{r} ,V}}^{•} (g^{′−1}M^{•}, g^{′−1}N^{•}) \ar[d]^{∼}_{(4.3.1)} & \underline{RHom}_{O_{F^{r} ,U}}^{•} (g^{!}f_{+}M^{•}, g^{!}f_{+}N^{•}) \ar[d]^{∼}_{( 3.9)} \\ Rf′_{∗}\underline{RHom}_{O_{F^{r} ,V}}^{•} (g^{′!}M^{•}, g^{′!}N^{•}) \ar[r]_{part (i)} & \underline{RHom}_{O_{F^{r} ,U}}^{•} (f′_{+}g^{′!}M^{•}, f′_{+}g^{′!}N^{•})}

(in which the morphisms are labelled the results which give rise to them) commutes. In other words, the natural transformation of (i) is local on the base.

(iii) Let f : Y → X and g : Z → Y be morphisms of smooth k-schemes. Then for any objects M^{•} and N^{•} of D_{qc}^{b}(O_{F^{r},Z}) the diagram of natural isomorphisms

\xymatrix{R(fg)_{∗}\underline{RHom}_{O_{F^{r} ,Z}}^{•} (M^{•}, N^{•}) \ar[d]^{∼} \ar[r]_{part (i)} & \underline{RHom}_{O_{F^{r} ,X}}^{•} ((fg)_{+}M^{•}, (fg)_{+}N^{•}) \ar[dd]^{∼}_{(3.8)} \\ Rf_{∗}Rg_{∗}\underline{RHom}_{O_{F^{r} ,Z}}^{•} (M^{•}, N^{•}) \ar[d]_{part (i)} \\ Rf_{∗}\underline{RHom}_{O_{F^{r} ,Y}}^{•} (g_{+}M^{•}, g_{+}N^{•}) \ar[r]_{part (i)} & \underline{RHom}_{O_{F^{r} ,X}}^{•} (f_{+}g_{+}M^{•}, f_{+}g_{+}N^{•})}

(in which the morphisms are labelled the results which give rise to them) commutes. In other words, the natural transformation of (i) is compatible with compositions.

Proof. Let us begin by proving (i). We may and do assume (via lemma 1.7.1 and corollary 3.5.13) that M^{•} is a complex bounded above of locally free left O_{F^{r},Y} - modules and that N^{•} is a bounded complex of left O_{F^{r},Y} -modules which are acyclic for the left-exact functor O_{F^{r},X←Y} ⊗_{O_{F^{r} ,Y}} –. Then we have natural morphisms

(4.4.3) \underline{RHom}_{O_{F^{r} ,Y}}^{•} (M^{•}, N^{•}) = \underline{Hom}_{O_{F^{r} ,Y}}^{•} (M^{•}, N^{•})

⟶^{f↦id⊗f} \underline{Hom}_{f^{−1}O_{F^{r} ,X}}^{•} (O_{F^{r},X←Y} ⊗_{O_{F^{r} ,Y}} M^{•}, O_{F^{r},X←Y} ⊗_{O_{F^{r} ,Y}} N^{•})

= \underline{Hom}_{f^{−1}O_{F^{r} ,X}}^{•} (O_{F^{r},X←Y} ⊗^{𝕃}_{O_{F^{r} ,Y}} M^{•}, O_{F^{r},X←Y} ⊗^{𝕃}_{O_{F^{r} ,Y}} N^{•})

⟶ \underline{RHom}_{f^{−1}O_{F^{r} ,X}}^{•} (O_{F^{r},X←Y} ⊗^{𝕃}_{O_{F^{r} ,Y}} M^{•}, O_{F^{r},X←Y} ⊗^{𝕃}_{O_{F^{r} ,Y}} N^{•}).

Let us denote the composite morphism by ϕ.

Next, we consider objects M′^{•} and N′^{•} of D^{b}(f^{−1}O_{F^{r},X}), and we may and do assume that M′^{•} is a bounded complex of flasque left f^{−1}O_{F^{r},X}-modules, and that N′^{•} is a complex, bounded below, of injective left f^{−1}(O_{F^{r},X})-modules. Then

\underline{RHom}_{f^{−1}O_{F^{r} ,X}}^{•} (M′^{•}, N′^{•}) = \underline{Hom}_{f^{−1}O_{F^{r} ,X}}^{•} (M′^{•}, N′^{•}).

Furthermore, since the members of N′^{•} are injective, this is a complex of flasque sheaves, and we get a natural morphism

(4.4.4) Rf_{∗}\underline{RHom}_{f^{−1}O_{F^{r} ,X}}^{•} (M′^{•}, N′^{•}) = f_{∗}\underline{Hom}_{f^{−1}O_{F^{r} ,X}}^{•} (M′^{•}, N′^{•})

→ \underline{Hom}_{O_{F^{r} ,X}}^{•} (f_{∗}M′^{•}, f_{∗}N′^{•}).

Since M′^{•} is a complex of flasque sheaves, the complex f_{∗}M′^{•} represents Rf_{∗}M′^{•}. Similarly, the complex f_{∗}N′^{•} represents Rf_{∗}N′^{•}. Furthermore, since f_{∗} is rightadjoint to the exact functor f^{−1}, we see that f_{∗}N′^{•} is a complex of left O_{F^{r},X}- injective modules. Thus

\underline{Hom}_{O_{F^{r} ,X}}^{•} (f_{∗}M′^{•}, f_{∗}N′^{•}) = \underline{RHom}_{O_{F^{r} ,X}}^{•} (Rf_{∗}M′^{•}, Rf_{∗}N′^{•}).

We define

ψ : \underline{RHom}_{f^{−1}O_{F^{r} ,X}}^{•} (M′^{•}, N′^{•}) → \underline{RHom}_{O_{F^{r} ,X}}^{•} (Rf_{∗}M′^{•}, Rf_{∗}N′^{•})

to be the morphism in the derived category corresponding to the morphism (4.4.4).

Finally, setting M′ = O_{F^{r},X←Y} ⊗^{𝕃}_{O_{F^{r} ,Y}} M^{•}, N′ = O_{F^{r},X←Y} ⊗^{𝕃}_{O_{F^{r} ,Y}} N^{•},

we take ψ ∘ Rf_{∗}ϕ for the required morphism. The fact that this morphism is independent of the choices made is standard homological algebra.

Part (ii) follows immediately from the fact that the construction is local on the base. We leave the verification of part (iii) (which is standard and tedious) to the reader. □

(4.4.5) In the following results we will have occasion to consider complexes of sheaves of (O_{F^{r}}, O_{F^{r}})-bimodules on smooth k-schemes. We will show that the derived categories of bounded complexes of such bimodules are stable under pushforwards.

Suppose that f : Y → X is a morphism of smooth k-schemes, and let M^{•} be a bounded complex of (O_{F^{r},Y} , O_{F^{r},Y})-bimodules. Using the finite length resolution of O_{F^{r},X←Y} by right-flat (f^{−1}O_{F^{r},X}, O_{F^{r},Y})-bimodules constructed in section 3.5 we see that the derived tensor product O_{F^{r},X←Y} ⊗_{O_{F^{r} ,Y}} M^{•} is naturally a complex of (f^{−1}O_{F^{r},X}, O_{F^{r},Y})-bimodules. The natural morphism

f^{−1}O_{F^{r},X} = f^{−1}O_{X}[F^{r}] → O_{Y} [F^{r}] = O_{F^{r},Y}

allows us to restrict scalars on the right from O_{F^{r},Y} to f^{−1}O_{F^{r},X} and thus to regard O_{F^{r},X←Y} ⊗_{O_{F^{r} ,Y}} M^{•} as a complex of (f^{−1}O_{F^{r},X}, f^{−1}O_{F^{r},X})-bimodules. Now if we resolve this complex by a complex of flasque (^{−1}O_{F^{r},X}, f^{−1}O_{F^{r},X})- bimodules, we find that f_{+}M^{•} = Rf_{∗}(O_{F^{r},X←Y} ⊗_{O_{F^{r} ,Y}} M^{•}) is naturally a complex of (O_{F^{r},X}, O_{F^{r},X})-bimodules.

For example, O_{F^{r},Y} is naturally an (O_{F^{r},Y} , O_{F^{r},Y})-bimodule, and the discussion of the preceding paragraph shows that f_{+}O_{F^{r},Y} is naturally an object of the bounded derived category of complexes of (O_{F^{r},X}, O_{F^{r},X})-bimodules. If we forget the right O_{F^{r},X}-module structure, then since O_{F^{r},Y} is induced by O_{Y} we see as in

the proof of 3.6.3 that f_{+}O_{F^{r},Y} ⟶^{∼} O_{F^{r},X} ⊗_{O_{X}} Rf_{∗}ω_{Y/X}.

(4.4.6) If M^{•} is a bounded complex of (O_{F^{r},X}, O_{F^{r},X})-bimodules, then since any object in D_{qc}^{b}(O_{F^{r},X}) has a resolution by a bounded above complex of locally free O_{F^{r},X}-modules, we may define the derived functor

f_{+}O_{F^{r},Y} ⊗^{𝕃}_{O_{F^{r} ,X}} – : D_{qc}^{b}(O_{F^{r},X}) → D^{−}(O_{F^{r},X}).

We say that M^{•} is good if the image of this functor actually lies in D_{qc}^{b}(O_{F^{r},X}). For example, if M^{•} = O_{F^{r},X} then this is just the identity functor, and so O_{F^{r},X} is certainly a good bimodule.

The following lemma, which is a variant for bimodules of proposition 4.2, shows among other things that the push-forward of a good complex of bimodules is again a good complex of bimodules.

Lemma 4.4.7. (i) Let f : Y → X be a morphism of smooth k-schemes and let M^{•} be a good bounded complex of (O_{F^{r},Y} , O_{F^{r},Y})-bimodules. If N^{•} is an object of D_{qc}^{b}(O_{F^{r},X}), then there is a natural isomorphism of objects of D^{b}(O_{F^{r},X}):

f_{+}(M^{•} ⊗^{𝕃}_{O_{F^{r} ,Y}} f^{!}N^{•}) ⟶^{∼} f_{+}M^{•}[d_{Y/X}] ⊗^{𝕃}_{O_{F^{r} ,X}} N^{•}.

In particular, f_{+}M^{•} is good.

(ii) Let f : Y → X be a morphism of smooth k-schemes and let g : U → X be an open immersion. Form the fibre product

\xymatrix{V \ar[r]^{g′} \ar[d]^{f′} & Y \ar[d]^{f} \\ U \ar[r]^{g} & X.}

Then for any good bounded complex M^{•} of (O_{F^{r},Y} , O_{F^{r},Y})-bimodules and any object N^{•} of D_{qc}^{b}(O_{F^{r},X}), g^{′!}M^{•} is also good, and the diagram of isomorphisms

\xymatrix{g^{!}(f_{+}M^{•}[d_{Y/X}] ⊗^{𝕃}_{O_{F^{r} ,X}} N^{•}) \ar[d]^{∼}_{(2.6)} \ar[r]^{∼}_{part (i)} & g^{!}f_{+}(M^{•} ⊗^{𝕃}_{O_{F^{r} ,Y}} f^{!}N^{•}) \ar[d]^{∼}_{(3.9)} \\ g^{!}f_{+}M^{•}[d_{Y/X}] ⊗^{𝕃}_{O_{F^{r} ,U}} g^{!}N^{•} \ar[d]^{∼}_{(3.9)} & f′_{+}g^{′!}(M^{•} ⊗^{𝕃}_{O_{F^{r} ,Y}} f^{!}N^{•}) \ar[d]^{∼}_{(2.6)} \\ f′_{+}g^{′!}M^{•}[d_{Y/X}] ⊗^{𝕃}_{O_{F^{r} ,U}} g^{!}N^{•} \ar[d]^{∼} \ar[d]^{∼}_{part (i)} & f′_{+}(g^{′!}M^{•} ⊗^{𝕃}_{O_{F^{r} ,V}} g^{′!}f^{!}N^{•}) \\ f′_{+}(g^{′!}M ⊗^{𝕃}_{O_{F^{r} ,V}} f^{′!}g^{!}N^{•}) \ar@{-->}[ur]^{∼}_{(2.4)}}

(in which the isomorphisms are labelled by the results which give rise to them) commutes. In other words, the natural isomorphism of (i) is compatible with localization on the base.

(iii) Let f : Y → X and g : Z → Y be morphisms of smooth k-schemes. Then for any good bounded complex of (O_{F^{r},Z}, O_{F^{r},Z})-bimodules and any object N^{•} of D_{qc}^{b}(O_{F^{r},X}), g_{+}M^{•} is also good (by part (i)) and the diagram of natural isomorphisms

\xymatrix{(fg)_{+}M^{•}[d_{Z/X}] ⊗^{𝕃}_{O_{F^{r} ,X}} N^{•} \ar[r]^{∼}_{part (i)} \ar[d]^{∼}_{(3.8)} & (fg)_{+}(M^{•} ⊗^{𝕃}_{O_{F^{r} ,Z}} (fg)^{!}N^{•}) \ar[dd]^{∼}_{(2.4), (3.8)} \\ f_{+}g_{+}M^{•}[d_{Z/X}] ⊗^{𝕃}_{O_{F^{r} ,X}} N^{•} \ar[d]^{∼}_{part (i)} \\ f_{+}(g_{+}M^{•}[d_{Z/Y}] ⊗^{𝕃}_{O_{F^{r} ,Y}} f^{!}N^{•}) \ar[r]^{∼}_{part (i)} & f_{+}g_{+}(M^{•} ⊗^{𝕃}_{O_{F^{r} ,Z}} g^{!}f^{!}N^{•})}

(in which the isomorphisms are labelled by the results which give rise to them) commutes. In other words, the natural isomorphism of (i) is compatible with composition.

Proof. To prove part (i) we may replace N^{•} by a complex bounded above of locally free O_{F^{r},X}-modules which has only finitely many non-vanishing cohomology sheaves. Let P^{•} be the finite length right-flat resolution of O_{F^{r},X←Y} constructed in section 3.5. Then

f_{+}(M^{•} ⊗^{𝕃}_{O_{F^{r} ,Y}} f^{!}N^{•}) = Rf_{∗}(P^{•} ⊗_{O_{F^{r} ,Y}} M^{•} ⊗_{f^{−1}O_{F^{r} ,X}} f^{−1}N^{•}[d_{Y/X}]).

Now the complex P^{•} ⊗_{O_{F^{r} ,Y}} M^{•} is of finite length, and the complex P^{•} ⊗_{O_{F^{r} ,Y}} M^{•} ⊗_{f^{−1}O_{F^{r} ,X}} f^{−1}N^{•}, while possibly unbounded below, has only finitely many non-vanishing cohomology sheaves (since we have assumed M^{•} to be good). Since Rf_{∗} has finite cohomological dimension, we can apply the projection formula to f and the complex of locally free O_{F^{r},X}-modules N^{•} to obtain

f_{+}(M^{•} ⊗^{𝕃}_{O_{F^{r} ,Y}} f^{!}N^{•}) ⟶^{∼} Rf_{∗}(P^{•} ⊗_{O_{F^{r} ,Y}} M^{•}) ⊗_{O_{F^{r} ,X}} N^{•}[d_{Y/X}]

= f_{+}M^{•}[d_{Y/X}] ⊗^{𝕃}_{O_{F^{r} ,X}} N^{•}.

This proves the formula of part (i). Since the left hand side lies in D_{qc}^{b}(O_{F^{r},X}) (by theorem 3.6.3 together with the assumption that M^{•} is good), we see that the same is true of the right hand side, and so f_{+}M^{•} is also good.

We now turn to proving part (ii). To see that g^{′!}M^{•} is good, first replace M^{•} by a complex bounded above of right-locally free bimodules, and then note that for any object N^{•} of D_{qc}^{b}(O_{F^{r},V}),

g^{!}M^{•} ⊗^{𝕃}_{O_{F^{r} ,V}} N^{•} = g^{!}M^{•} ⊗_{O_{F^{r} ,V}} N^{•}

= g^{−1}M^{•} ⊗_{O_{F^{r} ,V}} N^{•}

= g^{−1}(M^{•} ⊗_{O_{F^{r} ,Y}} Rg_{∗}N^{•})

= g^{!}(M^{•} ⊗^{𝕃}_{O_{F^{r} ,Y}} g_{+}N^{•}).

Since M is good, and both g_{+} and g^{!} preserve the property of being bounded with quasi-coherent cohomology sheaves, we see that g^{!}M^{•} is good. Now the commutative diagram of part (ii) follows immediately from the fact that the construction of the isomorphism of part (i) is local on the base (since the isomorphism provided by the projection formula is local on the base).

The proof of part (iii) is straightforward but tedious, so we leave it for the reader to verify. □

(4.4.8) Now suppose that f : Y → X is a proper morphism of smooth k-schemes. Then (as discussed in (A.1)) Grothendieck-Serre duality defines a canonical morphism of complexes of O_{X}-modules

Rf_{∗}ω_{Y/X}[d_{Y/X}] → O_{X},

which after tensoring on the left by O_{F^{r},X} over O_{X} yields a morphism of left O_{F^{r},X}-modules

f_{+}O_{F^{r},Y} [d_{Y/X}] ⟶^{∼} O_{F^{r},X} ⊗_{O_{X}} Rf_{∗}ω_{Y/X}[d_{Y/X}] → O_{F^{r},X}.

Now both the source and target of this morphism are naturally objects of the derived category of complexes of (O_{F^{r},X}, O_{F^{r},X})-bimodules, and we have the following proposition.

Proposition 4.4.9. (i) Let f : Y → X be a proper morphism of smooth k-schemes. Then there is a canonical morphism

tr_{F^{r},f} : f_{+}O_{F^{r},Y} [d_{Y/X}] → O_{F^{r},X}

in the derived category of complexes of (O_{F^{r},X}, O_{F^{r},X})-bimodules, which after forgetting the right O_{F^{r},X}-modules structure reduces to the morphism

O_{F^{r},X} ⊗_{O_{X}} Rf_{∗}ω_{Y/X}[d_{Y/X}] → O_{F^{r},X}

constructed via Grothendieck-Serre duality.

(ii) Let f : Y → X be a proper morphism of smooth k-schemes and let g : U → X be an open immersion. Form the fibre product

\xymatrix{V \ar[r]^{g′} \ar[d]^{f′} & Y \ar[d]^{f} \\ U \ar[r]^{g} & X.}

Then the diagram

\xymatrix{g^{!}f_{+}O_{F^{r},Y} [d_{Y/X}] \ar[d]^{∼}_{(3.9)} \ar[r]_{part (i)} & g^{!}O_{F^{r},X} \\ f′_{+}g^{′!}O_{F^{r},Y} [d_{Y/X}] = f′_{+}O_{F^{r},V} [d_{V/U}] \ar[ur]_{part (i)}}

(in which the morphisms are labelled by the result which gives rise to them) commutes. In other words, the construction of tr_{F^{r}} is compatible with localization on the base.

(iii) Let f : Y → X and g : Z → Y be proper morphisms of smooth k-schemes. Then the diagram

\xymatrix{(fg)_{+}O_{F^{r},Z}[d_{Z/X}] \ar[r]_{part (i)} \ar[d]^{∼}_{(3.8)} & O_{F^{r},X} \\ f_{+}g_{+}O_{F^{r},Z}[d_{Z/X}][d_{Y/X}] \ar[r]_{part (i)} & f_{+}O_{F^{r},Y} [d_{Y/X}] \ar[u]_{part (i)}}

(in which the morphisms are labelled by the result which gives rise to them) commutes. In other words, the formation of tr_{F^{r}} is compatible with composition.

Proof. We begin by proving part (i) of the proposition. Let us suspend for a moment the notation in the statement of the proposition, and let f : Y → X denote an arbitrary morphism of finite type k-schemes. Let us also suspend our usual notational convention concerning f^{!}, and use this to denote (not O_{F} -module pull-back, but rather) the functor which is normally so denoted in the theory of Grothendieck-Serre duality (as explained in [Ha] and [Con]). We use the theory of residual and pointwise dualizing complexes developed in [Ha, VI,VII]. In particular, recall that there is a functor f^{△} : Res(X) → Res(Y) from residual complexes on X to residual complexes on Y which realizes on the level of complexes (of quasicoherent, injective O_{X}-modules) the functor f^{!} restricted to pointwise dualizing complexes on X. Following [Ha], for a pointwise dualizing complex M^{•}, we denote by E^{•}(M^{•}) the unique residual complex which realizes it. It is pointed out in [Con] that E^{•} is in fact not functorial for arbitrary morphisms between pointwise dualizing complexes, and hence that f^{△} and the trace map tr_{f} are not functorial for arbitrary morphisms between residual complexes. However, Conrad shows that they are functorial for isomorphisms of pointwise dualizing complexes, and this is all that we need for the argument we are going to make.

Now let us return to the notation of the proposition, so that f : Y → X is a proper morphism of smooth k-schemes, and consider the r^{th} relative Frobenius diagram of f (see A.2). Since f is proper, the same is true of f^{(r)}. As explained in A.2, the morphism F_{Y/X}^{(r)} is finite, and so also proper. Thus the functors tr_{f}, tr_{f^{(r)}} and tr_{F_{Y /X}^{(r)}} all yield morphisms of complexes.

Note that the morphism F_{X}^{r} is residually stable (in the sense of [Ha], and so also in the sense of [Con]), meaning (following [Ha]) that it is flat and integral, with Gorenstein fibres. In fact, since X is smooth the fibres are even local complete intersections. It follows that the base-change F_{X}^{r′} is also residually stable. Thus F_{X}^{r∗}(E^{•}(O_{X})) is a residual complex, by [Ha, VI 5.3]. It provides a resolution of the pointwise dualizing complex F^{r∗}O_{X}, and [Con, I 10.2] shows that the the canonical isomorphism of pointwise dualising complexes F_{X}^{r∗}O_{X} ⟶^{∼} O_{X} is induced by a canonical isomorphism E^{•}(O_{X}) ⟶^{∼} F_{X}^{r∗}(E^{•}(O_{X})). Also, pulling back the residual complex f^{△}E^{•}(O_{X}) by the residually stable morphism F_{X}^{r′}, we see that F_{X}^{r′∗}f^{△}E^{•}(O_{X}) is a residual complex.

Consider the following diagram of morphisms of complexes of quasi-coherent O_{Y^{(r)}}-modules:

(4.4.10)  \xymatrix{F_{Y/X∗}^{(r)}f^{△}E^{•}(O_{X}) \ar[d]^{∼} \\ F_{Y/X∗}^{(r)}F_{Y/X}^{(r)△}f^{(r)△}E^{•}(O_{X}) \ar[d]^{∼} \ar[r]^{tr_{F{(r) \atop Y /X}} E^{•}(O_{X})} & f^{(r)△}E^{•}(O_{X}) \ar[d]^{∼} \\ F_{Y/X∗}^{(r)}F_{Y/X}^{(r)△}f^{(r)△}F_{X}^{r∗}E^{•}(O_{X}) \ar[d]^{∼} \ar[r]^{tr_{F{(r) \atop Y /X}} F_{X}^{r∗}E^{•}(O_{X})} & f^{(r)△}F_{X}^{r∗}E^{•}(O_{X}) \ar[d]^{∼} \\ F_{Y/X∗}^{(r)}F_{Y/X}^{(r)△}F_{X}^{r′∗}f^{△}E^{•}(O_{X}) \ar[r]^{tr_{F{(r) \atop Y /X}} F_{X}^{r′∗}f^{△}E^{•}(O_{X})} & F_{X}^{r′∗}f^{△}E^{•}(O_{X}).}

The top-most vertical arrow in this diagram is deduced from [Ha, VI 3.1] (compare [Con, I §10]). The upper square is defined using the isomorphism E^{•}(O_{X}) ⟶^{∼} F_{X}^{r∗}E^{•}(O_{X}) constructed above; that it commutes follows from the fact that tr_{F_{Y /X}^{(r)}}

is functorial for isomorphism of residual complexes. The lower square is constructed using residually stable base-change to obtain an isomorphism f^{(r)△}F_{X}^{r∗}E^{•}(O_{X}) ⟶^{∼} F_{X}^{r′∗}f^{△}E^{•}(O_{X}) [Ha, 5.5], and then applying functoriality of tr_{F_{Y /X}^{(r)}} to this isomor-

phism.

If we trace the maps in (4.4.10) through from the upper left to the lower right, we obtain a morphism F_{Y/X∗}^{(r)}f^{△}E^{•}(O_{X}) → F_{X}^{r′∗}f^{△}E^{•}(O_{X}), which (one sees after comparing with the definition given in A.2) provides a realization on the level of residual complexes of the relative Cartier operator C_{Y/X}^{(r)} : F_{Y/X∗}^{(r)}ω_{Y/X} → F_{X}^{r′∗}ω_{Y/X}, shifted by the amount d_{Y/X}.

Applying the induced bimodule construction of proposition-definition 1.11.1, we may endow the tensor product f^{−1}O_{F^{r},X} ⊗_{f^{−1}O_{X}} f^{(r)△}E^{•}(O_{X}) with the structure of a complex of (f^{−1}O_{F^{r},X}, O_{F^{r},Y})-bimodules, and (since f^{−1}O_{F^{r},X} is locally free as a right f^{−1}O_{X}-module) this complex provides a flasque resolution of the (f^{−1}O_{F^{r},X}, O_{F^{r},Y})-bimodule O_{F^{r},X←Y} [d_{Y/X}] = f^{−1}O_{F^{r},X} ⊗_{f^{−1}O_{X}} ω_{Y/X}[d_{Y/X}]. Thus we may apply f_{∗} to this resolution to compute the complex of (O_{F^{r},X}, O_{F^{r},X})- bimodules f_{+}O_{F^{r},Y}.

In order to continue our construction, we apply f_{∗}^{(r)} to diagram (4.4.10), and then embed the result in the following larger commutative diagram (in which we have abbreviated E^{•}(O_{X}) to E^{•}):

(4.4.11) [figure: f_{∗}f^{△}E^{•}; tr_{f} E^{•}; ∼; E^{•}; f_{∗}^{(r)}F_{Y/X∗}^{(r)}f^{△}E^{•}; ∼; f_{∗}^{(r)}F_{Y/X∗}^{(r)}F_{Y/X}^{(r)△}f^{(r)△}E^{•}; ∼; f_{∗}^{(r)}tr_{F{(r) \atop Y /X}} E^{•}; f_{∗}^{(r)}f^{(r)△}E^{•}; ∼; tr_{f^{(r)}} E^{•}; E^{•}; ∼; f_{∗}^{(r)}F_{Y/X∗}^{(r)}F_{Y/X}^{(r)△}f^{(r)△}F_{X}^{r∗}E^{•}; ∼; f_{∗}^{(r)}tr_{F{(r) \atop Y /X}} F_{X}^{r∗}E^{•}; f_{∗}^{(r)}f^{(r)△}F_{X}^{r∗}E^{•}; ∼; tr_{f^{(r)}}; F_{X}^{r∗}E^{•}; f_{∗}^{(r)}F_{Y/X∗}^{(r)}F_{Y/X}^{(r)△}F_{X}^{r′∗}f^{△}E^{•}; f_{∗}^{(r)}tr_{F{(r) \atop Y /X}} F_{X}^{r′∗}f^{△}E^{•}; f_{∗}^{(r)}F_{X}^{r′∗}f^{△}E^{•}; ∼; F_{X}^{r∗}f_{∗}f^{△}E^{•}; F_{X}^{r∗}tr_{f}; F_{X}^{r∗}E^{•}.]

The top-most rectangle of this diagram is constructed via [Ha, VI 4.2 TRA 1)]. The centre right rectangle is constructed by applying functoriality of tr_{f} to the isomorphism E^{•}(O_{X}) ⟶^{∼} F_{X}^{r∗}E^{•}(O_{X}). The lower right rectangle is constructed using residually stable base-change, and its commutativity follows from [Ha, VI 5.6]. The remainder of the diagram is obtained by applying f_{∗}^{(r)} to (4.4.10).

From this diagram we can extract the following commutative rectangle:

\xymatrix{f_{∗}f^{△}E^{•}(O_{X}) \ar[d] \ar[r]^{tr_{f} E^{•}(O_{X})} & E^{•}(O_{X}) \ar[d] \\ F_{X}^{r∗}f_{∗}f^{△}E^{•}(O_{X}) \ar[r]^{F_{X}^{r∗}tr_{f} E^{•}(O_{X})} & F_{X}^{r∗}E^{•}(O_{X}).}

By (1.11.4) and lemma 1.11.2 we see that each of O_{F^{r},X} ⊗_{O_{X}} f_{∗}f^{△}E^{•}(O_{X}) and O_{F^{r},X} ⊗_{O_{X}} E^{•}(O_{X}) is endowed with the structure of an (O_{F^{r},X}, O_{F^{r},X})-bimodule, and that tr_{f}E^{•}(O_{X}) induces a morphism of bimodules

tr_{F^{r},f} : O_{F^{r},X} ⊗_{O_{X}} f_{∗}f^{△}E^{•}(O_{X}) → O_{F^{r},X} ⊗_{O_{X}} E^{•}(O_{X}).

Furthermore, (1.11.5) shows that the bimodule structure that we have induced on O_{F^{r},X} ⊗_{O_{X}} f_{∗}f^{△}E^{•}(O_{X}) is identical to the bimodule structure obtained by pushing forward the (f^{−1}O_{F^{r},X}, O_{F^{r},Y})-bimodule structure on f^{−1}O_{F^{r},X} ⊗_{f^{−1}O_{X}} f^{(r)△}E^{•}(O_{X}), and so O_{F^{r},X} ⊗_{O_{X}} f_{∗}f^{△}E^{•}(O_{X}) is a complex of (O_{F^{r},X}, O_{F^{r},X})- bimodules which represents f_{+}O_{F^{r},Y}. On the other hand, since E^{•}(O_{X}) is a resolution of O_{X}, and since O_{F^{r},X} is flat as a right O_{X}-module, we see that O_{F^{r},X} ⊗_{O_{X}} E^{•}(O_{X}) is a resolution of O_{F^{r},X} as a complex of (O_{F^{r},X}, O_{F^{r},X})- bimodules. Putting this all together, we see we have indeed found the desired morphism

tr_{F^{r},f} : f_{+}O_{F^{r},Y} [d_{Y/X}] → O_{F^{r},X}.

Since all of our constructions may be localized on X, part (ii) of the proposition is immediate. The proof of part (iii) relies on the compatibility of the trace maps of coherent duality with composition [Ha, VI 4.2]. Granting this, the necessary verification is standard but tedious, and is left to the reader. □

(4.4.12) Proof of 4.4.1. Let f : Y → X be a proper morphism of smooth, k-schemes. Let M^{•} be be an object of D_{qc}^{b}(O_{F^{r},Y}) and let N^{•} be an object of D_{qc}^{b}(O_{F^{r},X}). Combining the results of parts (i) of propositions 4.4.2 and 4.4.9 and lemma 4.4.7 we obtain a natural transformation

Rf_{∗}\underline{RHom}_{O_{F^{r} ,Y}}^{•} (M^{•}, f^{!}N^{•}) ⟶ \underline{RHom}_{O_{F^{r} ,X}}^{•} (f_{+}M^{•}, f_{+}f^{!}N^{•})

⟶^{∼} \underline{RHom}_{O_{F^{r} ,X}}^{•} (f_{+}M^{•}, f_{+}O_{F^{r},Y} [d_{Y/X}] ⊗^{𝕃}_{O_{F^{r} ,X}} N^{•})

⟶ \underline{RHom}_{O_{F^{r} ,X}}^{•} (f_{+}M^{•}, O_{F^{r},X} ⊗^{𝕃}_{O_{F^{r} ,X}} N^{•})

= \underline{RHom}_{O_{F^{r} ,X}}^{•} (f_{+}M^{•}, N^{•}).

It remains to show that this is an isomorphism.

For this, we replace M^{•} by a complex, bounded above, of induced quasi-coherent left O_{F^{r},Y} -modules, and by the usual spectral sequence argument, it suffices to verify that the natural transformation is indeed an isomorphism for the case of a single induced quasi-coherent left O_{F^{r},X}-module M = O_{F^{r},X} ⊗_{O_{X}} M (where M is a quasi-coherent O_{X}-module).

In this case

f_{+}M = O_{F^{r},X} ⊗_{O_{X}} Rf_{∗}(ω_{Y/X} ⊗_{O_{Y}} M),

and so there are natural isomorphisms

\underline{RHom}_{O_{X}}^{•} (M, f^{!}N^{•}) ⟶^{∼} \underline{RHom}_{O_{F^{r} ,X}}^{•} (M, f^{!}N^{•})

and

\underline{RHom}_{O_{X}}^{•} (Rf_{∗}(ω_{Y/X} ⊗_{O_{Y}} M), N^{•}) ⟶^{∼} \underline{RHom}_{O_{F^{r} ,X}}^{•} (f_{+}M, N^{•}).

Finally we have natural maps

Rf_{∗}\underline{RHom}_{O_{X}}^{•} (M, f^{!}N^{•})

⟶ \underline{RHom}_{O_{X}}^{•} (Rf_{∗}(ω_{Y/X} ⊗_{O_{X}} M), Rf_{∗}ω_{Y/X}[d_{Y/X}] ⊗^{𝕃}_{O_{Y}} N^{•})

⟶^{∼} \underline{RHom}_{O_{X}}^{•} (Rf_{∗}(ω_{Y/X} ⊗_{O_{Y}} M), N^{•}),

and the composite is an isomorphism by Grothendieck-Serre duality. (Note that here we are using the “explicit trace map” form of duality, rather than that based on dualizing complexes, and that we are also applying duality to the (not necessarily coherent) quasi-coherent complex N^{•}. The justification for this is provided in (A.1).) These isomorphisms fit into the following commutative diagram:

Rf_{∗}\underline{RHom}_{O_{X}}^{•} (M, f^{!}N^{•})  ⟶^{∼} Rf_{∗}\underline{RHom}_{O_{F^{r} ,X}}^{•} (M, f^{!}N^{•})
∼↓  ↓
\underline{RHom}_{O_{X}}^{•} (Rf_{∗}(ω_{Y/X} ⊗_{O_{Y}} M), N^{•}) ⟶^{∼} \underline{RHom}_{O_{F^{r} ,X}}^{•} (f_{+}M, N^{•}),

So the right hand vertical arrow is an isomorphism, and part (i) of the theorem is proved.

Parts (ii) and (iii) of the isomorphism follow from the construction of the adjunction map, and the corresponding parts of propositions 4.4.2 and 4.4.9 and lemma 4.4.7. □

(4.4.13) It will be useful to make proposition 4.4.9 explicit in the case when f : Y → X is the closed immersion of a smooth divisor into a smooth k-scheme X.

If a is a local equation for Y, we can represent the commutative rectangle

\xymatrix{f_{∗}f^{△} \ar[r]^{tr_{f} E^{•}(O_{X})} \ar[d] & E^{•}(O_{X}) \ar[d]^{∼} \\ F_{X}^{r∗}f_{∗}f^{△}E^{•}(O_{X}) \ar[r]^{F_{X}^{r∗}tr_{f} E^{•}(O_{X})} & F_{X}^{r∗}E^{•}(O_{X})}

in a more concrete fashion via Koszul complexes. Following the sign conventions of [Con, I §1] the resulting diagram (in which the upper left complex is K^{•}(a, O_{X}) and the lower left complex is K^{•}(a^{p}, O_{X})) is

[figure: (O_{X}; ·−a; O_{X}); ·a^{p−1}; (O_{X}; 0); (O_{X}; ·−a^{p}; O_{X}); (O_{X}; 0).]

Applying the induced bimodule construction of 1.11 we obtain a morphism of complexes of (O_{F^{r},X}, O_{F^{r},X})-bimodules

(O_{F^{r},X} → O′_{F^{r},X}) → O_{F^{r},X}

representing the morphism f_{+}O_{F^{r},Y} [d_{Y/X}] → O_{F^{r},X}; here O′_{F^{r},X} denotes O_{F^{r},X} with its usual left O_{F^{r},X}-module structure, but with its right module structure defined by

F^{rn} · F^{r} = a^{q^{n}(q−1)}F^{r(n+1)}.

The differential O_{F^{r},X} → O′_{F^{r},X} is defined by F^{rn} ↦ a^{q^{n}}F^{rn}.

It will be useful to note for our applications that (since O′_{F^{r},X} is isomorphic to O_{F^{r},X} as a right O_{X}-module) both O_{F^{r},X} and O′_{F^{r},X} are acyclic with respect to tensor product on the right over O_{F^{r},X} by induced left O_{F^{r},X}-modules.

(4.5) Let f : Y → X be a closed immersion of k-schemes, and let I be the ideal cutting out Y in X. Since f is proper, theorem 4.4.1 yields for any complex M^{•} in D_{qc}^{b}(O_{F^{r},Y}) an adjunction morphism f_{+}f^{!}M^{•} → M^{•}. Now the source of this arrow has the following rather explicit description (if one combines corollary 3.5.8 with proposition 2.8.4): it is equal to

f_{∗}(f^{−1}O_{F^{r},X} ⊗_{f^{−1}O_{X}} ω_{Y/X} ⊗_{O_{F^{r} ,Y}} ω_{Y/X}^{−1} ⊗_{O_{Y}} \underline{RHom}_{O_{X}}^{•} (f_{∗}O_{Y} , M^{•})).

There is an obvious morphism

(4.5.1) f_{∗}(f^{−1}O_{F^{r},X} ⊗_{f^{−1}O_{X}} ω_{Y/X} ⊗_{O_{Y}} ω_{Y/X}^{−1} ⊗_{O_{Y}} \underline{RHom}_{O_{X}}^{•} (f_{∗}O_{Y} , M^{•}))

⟶ M^{•},

given as follows: if I^{•} is a bounded below complex of injective left O_{F^{r},X}-modules resolving M^{•}, then on the level of complexes (4.5.1) is simply the morphism

f_{∗}(f^{−1}O_{F^{r},X} ⊗_{f^{−1}O_{X}} ω_{Y/X} ⊗_{O_{Y}} ω_{Y/X}^{−1} ⊗_{O_{Y}} I^{•}[I])

⟶^{∼} O_{F^{r},X} ⊗_{O_{X}} f_{∗}(I^{•}[I]) ⟶ I^{•}

given by the projection formula together with multiplication of O_{F^{r},X} on the members of I^{•}. There is also a natural morphism

(4.5.2) f_{∗}(f^{−1}O_{F^{r},X} ⊗_{f^{−1}O_{X}} ω_{Y/X} ⊗_{O_{Y}} ω_{Y/X}^{−1} ⊗_{O_{Y}} \underline{RHom}_{O_{X}}^{•} (f_{∗}O_{Y} , M^{•}))

⟶ f_{∗}(f^{−1}O_{F^{r},X} ⊗_{f^{−1}O_{X}} ω_{Y/X} ⊗_{O_{F^{r} ,Y}} ω_{Y/X}^{−1} ⊗_{O_{Y}} \underline{RHom}_{O_{X}}^{•} (f_{∗}O_{Y} , M^{•})),

given by contracting the tensor product over O_{Y} to a tensor product over O_{F^{r},Y}.

The following result is included to provide a comparison of our (somewhat abstract) constructions with the concrete constructions of [Lyu, Prop. 3.1].

Proposition 4.5.3. Let f : Y → X be a closed immersion of smooth k-schemes, and let M^{•} be a complex in D_{qc}^{b}(O_{F^{r},X}). Then the composite of the adjunction morphism f_{+}f^{!}M^{•} → M^{•} with the morphism (4.5.2) is equal to the morphism (4.5.1).

Proof. To prove this one has to reconcile the isomorphism of proposition 2.8.4 with the proof of proposition 4.4.9. Let E^{•} = E^{•}(O_{X}) be the residual complex resolving O_{X}. Since X is regular of dimension d_{X}, E^{•} is a complex of length d_{X}. Then E^{•}[I] is the complex f^{△}E^{•}, and the map tr_{f}E^{•} is simply the inclusion f_{∗}(E^{•}[I]) → E^{•}. The fundamental local isomorphism (discussed in (A.1.3)) shows that E^{•}[I] is a left resolution of ω_{Y/X}[d_{Y/X}].

Let us recall the construction of the r^{th} relative Cartier operator in this context, following the proof of proposition 4.4.9. The isomorphism O_{X} ⟶^{∼} F_{X}^{r∗}O_{X} induces an isomorphism of complexes E^{•} ⟶^{∼} F_{X}^{r∗}E^{•}, which in turn induces an isomorphism

E^{•}[I^{(r)}] ⟶^{∼} (F_{X}^{r∗}E^{•})[I^{(r)}] ⟶^{∼} F_{X}^{r′∗}(E^{•}[I]).

Composing this with the map F_{Y/X∗}^{(r)}E^{•}[I] → E^{•}[I^{(r)}] we obtain the map

F_{Y/X∗}^{(r)}E^{•}[I] → F_{X}^{r′∗}(E^{•}[I])

which realizes the relative Cartier operator (shifted by d_{Y/X}) on the level of residual complexes. This morphism gives f^{−1}O_{F^{r},X}⊗_{f^{−1}O_{X}} E^{•}[I] the structure of a complex of induced (f^{−1}O_{F^{r},X}, O_{F^{r},Y})-bimodules, resolving O_{F^{r},X←Y} [d_{Y/X}]

There is one fact which we did not take notice of in the proof of 4.4.9, but that is useful to note here. Namely, the inverse of the above isomorphism E^{•} → F_{X}^{r∗}E^{•} provides a structural morphism for E^{•}, making it a complex of left O_{F^{r},X}-modules which resolves O_{X} as a left O_{F^{r},X}-module. (This a generalization to the non-affine case of the observation of [Lyu, Ex. 1.2 (b”)].) The construction of corollary 2.8.2 then applies to give ω_{Y/X}^{−1} ⊗_{O_{Y}} E^{•}[I] the structure of a complex of left O_{F^{r},Y} - modules. This complex is a left resolution of ω_{Y/X}^{−1} ⊗_{O_{Y}} ω_{Y/X} ⟶^{∼} O_{Y}. It follows from the construction of 2.8.2 that the the augmentation

(4.5.4)  ω_{Y/X}^{−1} ⊗_{O_{Y}} E^{•}[I] → O_{Y} [d_{Y/X}]

is a morphism of left O_{F^{r},Y} -modules.

We may and do assume that M^{•} is a bounded above complex of locally free left O_{F^{r},X}-modules. Then the members of the complex M^{•} are in particular locally free as O_{X}-modules. Thus E^{•} ⊗_{O_{X}} M^{•} is a resolution of M^{•} by a bounded below complex of left O_{F^{r},X}-modules (here we are using the left O_{F^{r},X}-module structure on the complex E^{•} to put a left O_{F^{r},X}-module structure on the tensor product) which are injective as O_{X}-modules. (To see this, note that X is noetherian, so that a direct limit of injective O_{X}-modules is injective and the property of being injective as an O_{X}-module may be checked locally.) Thus

\underline{RHom}_{O_{X}}^{•} (f_{∗}O_{Y} , M^{•}) ⟶^{∼} (E^{•} ⊗_{O_{X}} M^{•})[I] ⟶^{∼} E^{•}[I] ⊗_{O_{Y}} f^{∗}M^{•},

where the second isomorphism follows from the local freeness of the members of M^{•} as O_{X}-modules. Note that this is a morphism of complexes of O_{Y} -modules; neither side has a natural left O_{F^{r},Y} -module structure. However, we may tensor both sides of this isomorphism by ω_{Y/X}^{−1} over O_{Y} to obtain a morphism

(4.5.5)  ω_{Y/X}^{−1} ⊗_{O_{Y}} (E^{•} ⊗_{O_{X}} M^{•})[I] ⟶^{∼} ω_{Y/X}^{−1} ⊗_{O_{Y}} E^{•}[I] ⊗_{O_{Y}} f^{∗}M.

Now both sides of this isomorphism are naturally complexes of left O_{F^{r},Y} -modules, the left side by applying corollary 2.8.2 to E^{•} ⊗_{O_{X}} M^{•}, and the right side by first applying corollary 2.8.2 to E^{•} as above, and then tensoring the resulting complex of left O_{F^{r},Y} -modules over O_{Y} with the complex of left O_{F^{r},Y} -modules f^{∗}M^{•}. It is clear from the construction of corollary 2.8.2 that (4.5.5) is an isomorphism of complexes of left O_{F^{r},Y} -modules.

The quasi-isomorphism (4.5.4) yields a quasi-isomorphism of complexes of left O_{F^{r},Y} -modules

ω_{Y/X}^{−1} ⊗_{O_{Y}} E^{•}[I] ⊗ f^{∗}M^{•} → O_{Y} [d_{Y/X}] ⊗_{O_{Y}} f^{∗}M^{•} ⟶^{∼} f^{∗}M^{•}[d_{Y/X}],

which when composed with (4.5.5) yields a quasi-isomorphism

ω_{Y/X} ⊗_{O_{Y}} (E^{•} ⊗_{O_{X}} M^{•})[I] → f^{∗}M^{•}[d_{Y/X}].

This is a realization on the level of complexes of the isomorphism of proposition 2.8.4 (as one sees by examining the construction of [Ha, I 7.4] and [Con, I 2.1]).

There are three more morphisms that we will need to consider. First, taking the tensor product over f^{−1}O_{X} with f^{−1}O_{F^{r},X} of the quasi-isomorphism

E^{•}[I][d_{Y/X}] → ω_{Y/X}

yields a quasi-isomorphism of complexes of induced (f^{−1}O_{F^{r},X}, O_{F^{r},Y})-bimodules

(4.5.6)  f^{−1}O_{F^{r},X} ⊗_{f^{−1}O_{X}} E^{•}[I][d_{Y/X}] → f^{−1}O_{F^{r},X} ⊗_{f^{−1}O_{X}} ω_{Y/X}.

The target of this quasi-isomorphism is O_{F^{r},X←Y} . We will denote the source by O′_{F^{r},Y →X}, for ease of type-setting. Second, the isomorphism

ω_{Y/X} ⊗_{O_{Y}} ω_{Y/X}^{−1} ⟶^{∼} O_{Y}

induces an isomorphism of f^{−1}O_{F^{r},X}-modules

(4.5.7) O_{F^{r},X←Y} ⊗_{O_{Y}} ω_{Y/X} = f^{−1}O_{F^{r},X} ⊗_{f^{−1}O_{X}} ω_{Y/X} ⊗_{O_{Y}} ω_{Y/X}

⟶^{∼} f^{−1}O_{F^{r},X} ⊗_{f^{−1}O_{X}} O_{Y} .

The final morphism we wish to consider is the morphism

(4.5.8) f^{−1}O_{F^{r},X} ⊗_{f^{−1}O_{X}} (E^{•} ⊗_{O_{X}} M^{•})[I]

⟶^{∼} f^{−1}O_{F^{r},X} ⊗_{f^{−1}O_{X}} E^{•}[I] ⊗_{O_{Y}} f^{∗}M^{•}

⟶ f^{−1}O_{F^{r},X} ⊗_{f^{−1}O_{X}} E^{•}[I] ⊗_{O_{F^{r} ,Y}} f^{∗}M^{•}

⟶^{∼} O′_{F^{r},X←Y} ⊗_{O_{F^{r} ,Y}} f^{∗}M^{•}[d_{Y/X}].

induced by the morphism (4.5.5) together with the contraction of the tensor product over O_{Y} to a tensor product over O_{F^{r},Y}.

We now put these morphisms together into the following diagram of morphisms of complexes of left O_{F^{r},X}-modules (in which the non-obvious maps have been labelled by the result giving rise to them, and the maps labelled q.i. are quasiisomorphisms):

\xymatrix{f_{∗}(O′_{F^{r},X←Y} ⊗_{O_{Y}} ω_{Y/X}^{−1} ⊗_{O_{Y}} (E^{•} ⊗_{O_{X}} M^{•})[I]) \ar[dd]^{q.i.}_{(4.5.6)} \ar@{-->}[dr]_{(4.5.5)} \\  & f_{∗}(O′_{F^{r},X←Y} ⊗_{O_{Y}} ω_{Y/X}^{−1} ⊗_{O_{Y}} E^{•}[I] ⊗_{O_{Y}} f^{∗}M^{•}) \ar[dd]^{q.i.}_{(4.5.4)} \\ f_{∗}(O_{F^{r},X←Y} ⊗_{O_{Y}} ω_{Y/X}^{−1} ⊗_{O_{Y}} (E^{•} ⊗ M^{•})[I]) \ar[dd]^{∼}_{(4.5.7)} \\  & f_{∗}(O′_{F^{r},X←Y} ⊗_{O_{F^{r} ,Y}} f^{∗}M^{•}[d_{Y/X}]) \ar[dd]^{∼} \\ f_{∗}(f^{−1}O_{F^{r},X} ⊗_{f^{−1}O_{X}} (E^{•} ⊗_{O_{X}} M^{•})[I]) \ar@{-->}[ur]_{(4.5.8)} \ar[dd]^{∼} \\  & O_{F^{r},X} ⊗_{O_{X}} f_{∗}(E^{•}[I]) ⊗_{O_{F^{r} ,X}} M^{•} \ar[dd] \\ O_{F^{r},X} ⊗_{O_{X}} f_{∗}((E^{•} ⊗_{O_{X}} M^{•})[I]) \ar[d]^{∼} \ar[ur] \\ O_{F^{r},X} ⊗_{O_{X}} E^{•} ⊗_{O_{X}} M^{•} \ar[r] \ar[ddr] & O_{F^{r},X} ⊗_{O_{X}} E^{•} ⊗_{O_{F^{r} ,X}} M^{•} \ar[d]^{∼}_{(1.11.8)} \\  & E^{•} ⊗_{O_{X}} O_{F^{r},X} ⊗_{O_{F^{r} ,X}} M^{•} \ar[ddr]^{∼} \\  & O_{F^{r},X} ⊗_{O_{F^{r} ,X}} E^{•} ⊗_{O_{X}} M^{•} \ar[dr]^{∼} \\  &  & E^{•} ⊗_{O_{X}} M^{•} \\  &  & M^{•} \ar[u]^{q.i.} \\ O_{F^{r},X} ⊗_{O_{X}} M^{•} \ar[r] \ar[uuuuu]^{q.i.} & O_{F^{r},X} ⊗_{O_{F^{r} ,X}} M^{•}. \ar[uuu]^{q.i.} \ar@{-->}[ur]^{∼}}

It is clear that every part of this diagram other than the top pentagon is a commuting diagram of morphisms of complexes of O_{F^{r},X}-modules. The top pentagon need not commute in the category of complexes of left O_{F^{r},X}-modules, but if we take the two morphisms from f_{∗}(O′_{F^{r},X←Y} ⊗_{O_{Y}} ω_{Y/X}^{−1} ⊗_{O_{Y}} (E^{•} ⊗_{O_{X}} M^{•})[I]) to f_{∗}(O_{F^{r},X←Y} ⊗_{O_{F^{r} ,Y}} f^{∗}M^{•}[d_{Y/X}]) obtained by going either way around this pentagon, and compose them with the quasi-isomorphism

f_{∗}(O′_{F^{r},X←Y} ⊗_{O_{F^{r} ,Y}} f^{∗}M^{•}[d_{Y/X}]) → f_{∗}(O_{F^{r},X←Y} ⊗_{O_{F^{r} ,Y}} f^{∗}M^{•}[d_{Y/X}])

the resulting two morphisms are equal. (This follows from the associativity of the tensor product, applied to the isomorphism

ω_{Y/X} ⊗_{O_{Y}} ω_{Y/X}^{−1} ⊗_{O_{Y}} ω_{Y/X} ⟶^{∼} ω_{Y/X}.)

Thus this pentagon does commute after passing to the derived category, and this diagram yields a commutative diagram in the derived category D_{qc}^{b}(O_{F^{r},X}). One immediately sees that the resulting diagram simplifies to give the diagram in the statement of the proposition. □

Remark 4.6 Let us note that all the constructions of this section are compatible with change of ground field via an algebraic extension k′/k, in the obvious sense. Indeed, the only subtle point is the construction of proposition 4.4.9, which involves base-change of the trace map and residual complexes. However, an algebraic field extension is residually stable, and base-change via residually stable morphisms is compatible with the formation of residual complexes and the trace map.

5. unit O_{F^{r}}-modules

Definition 5.1. A unit O_{F^{r},X}-module on a smooth k-scheme X is a quasi-coherent left O_{F^{r},X}-module M for which the structural morphism ϕ_{M} : F_{X}^{r}^{∗}M → M is an isomorphism of O_{X}-modules.

Lemma 5.2. Suppose that

M_{•} = M_{1} → M_{2} → M_{3} → M_{4} → M_{5}

is an exact sequence of left O_{F^{r},X}-modules, and that M_{1}, M_{2}, M_{4}, and M_{5} are unit O_{F^{r},X}-modules. Then M_{3} is a unit O_{F^{r},X}-module.

Proof. Since the M_{i} are quasi-coherent for i ≠ 3, we conclude that M_{3} is quasicoherent. Since X is smooth, F_{X}^{r}^{∗} is exact, and we get a morphism of exact sequences F_{X}^{r∗}M_{•} ⟶ M_{•}, so that the result follows from the five lemma. □

(5.3) The notion of unit O_{F^{r},X}-module, when r = 1 and X = Spec A is an affine scheme, is identical to that of F -module as defined in [Lyu]. Lyubeznik gives a method for constructing F -modules via a generating morphism, which immediately generalizes to the case of general r and an arbitrary smooth k-scheme X.

Construction-Definition 5.3.1. Suppose that M is a quasi-coherent O_{X}-module equipped with a morphism of O_{X}-modules

β : M → F_{X}^{r}^{∗}M.

We define M to be the direct limit of the direct system

M →^{β} F_{X}^{r}^{∗}M →^{F_{X}^{r}^{∗}β} (F_{X}^{2r})^{∗}M →^{(F_{X}^{2r})^{∗}β} ⋯ →^{(F_{X}^{rn−r})^{∗}β} (F_{X}^{rn})^{∗}M →^{(F_{X}^{rn})^{∗}β} · · · .

Since pull-back commutes with direct limits, F_{X}^{r∗}M is the direct limit of the system

F_{X}^{r∗}M →^{F_{X}^{r∗}β} (F_{X}^{2r})^{∗}M →^{(F_{X}^{2r})^{∗}β} ⋯ →^{(F_{X}^{rn−r})^{∗}β} (F_{X}^{rn})^{∗}M →^{(F_{X}^{rn})^{∗}β} ⋯,

and so is naturally identified with M. This identification F_{X}^{r}^{∗}M = M gives M the structure of unit O_{F^{r},X}-module (by lemma 1.5.2).

We say that M is the unit O_{F^{r},X}-module generated by the morphism β : M → F_{X}^{r}^{∗}M. More generally, if N is a unit O_{F^{r},X}-module isomorphic to M, we will say that N is generated by β, or that β is a generator of N.

(5.3.2) If M is a unit O_{F^{r},X}-module, let β denote the inverse to the structural morphism:

β = ϕ_{M}^{−1} : M → F_{X}^{∗}M.

Then the unit O_{F^{r},X}-module generated by β is naturally isomorphic to M. Thus any unit O_{F^{r},X}-module has at least one generator.

The following result gives an alternative description of the unit O_{F^{r},X}-module generated by a morphism β : M → F_{X}^{r}^{∗}M:

Proposition 5.3.3. Let M be an O_{X}-module equipped with an O_{X}-linear map β : M → F_{X}^{r}^{∗}M. Let β′ be the corresponding morphism of induced modules

β′ : O_{F^{r},X} ⊗_{O_{X}} M → O_{F^{r},X} ⊗_{O_{X}} M

(in the sense of (1.10.3)); equivalently, β′ can be described as right multiplication by F^{r} on O_{F^{r},X} ⊗_{O_{X}} M, when this tensor product is given its induced bimodule structure via the discussion of (1.11.4). Let M be the unit O_{F^{r},X}-module generated by β. Then M sits in the following short exact sequence:

0 ⟶ O_{F^{r},X} ⊗_{O_{X}} M ⟶^{1−β′} O_{F^{r},X} ⊗_{O_{X}} M ⟶ M ⟶ 0.

Proof. The natural map of left O_{X}-modules M → M induces a map of left O_{F^{r},X}- modules O_{F^{r},X} ⊗_{O_{X}} M → M, which is surjective by definition (see 1.5).

Furthermore, from the definition of the map β′ and the construction of the direct limit we now see that M is presented as

O_{F^{r},X} ⊗_{O_{X}} M →^{1−β′} O_{F^{r},X} ⊗_{O_{X}} M → M → 0.

It remains to see that the first arrow is injective. But this is clear, since β′ has image in O_{F^{r},X}F^{r} ⊗_{O_{X}} M, so that any element in the kernel of 1 − β′ lies in the intersection ⋂_{n=0}^{∞} O_{F^{r},X}F^{rn} ⊗_{O_{X}} M = 0. □

(5.4) It is an immediate consequence of lemma 1.9.2 that if M and N are unit O_{F^{r},X}-modules then M ⊗_{O_{X}} N is a unit O_{F^{r},X}-module. The following lemma makes an even stronger observation:

Lemma 5.4.1. Suppose that β : M → F_{X}^{r}^{∗}M generates the unit O_{F^{r},X}-module M and that γ : N → F_{X}^{r}^{∗}N generates the unit O_{F^{r},X}-module N . Then the tensor product

β ⊗ γ : M ⊗_{O_{X}} N → F_{X}^{r}^{∗}M ⊗_{O_{X}} F_{X}^{r}^{∗}N = F_{X}^{r}^{∗}(M ⊗_{O_{X}} N)

(using the isomorphism of lemma 1.9.1) generates the tensor product M ⊗_{O_{X}} N .

Proof. As ⊗_{O_{X}} commutes both with direct limits and with (F_{X}^{rn})^{∗}, this is clear. □

Definition 5.5. We define μ_{u}(X) to be the full subcategory of μ(X) consisting of unit O_{F^{r},X}-modules. We define D_{u}^{b}(O_{F^{r},X}) to be the full triangulated subcategory of D_{qc}^{b}(O_{F^{r},X}) consisting of complexes whose cohomology sheaves are unit O_{F^{r},X}- modules.

Note that lemma 5.2 shows that μ_{u}(X) is a thick subcategory of μ(X), so that D_{u}^{b}(O_{F^{r},X}) is indeed a triangulated sub-category of D_{qc}^{b}(O_{F^{r},X}).

Lemma 5.5.1. A complex M^{•} of D_{qc}^{b}(O_{F^{r},X}) belongs to D_{u}^{b}(O_{F^{r},X}) if and only if the structural isomorphism ϕ_{M^{•}} : F^{r∗}M^{•} → M^{•} is an isomorphism of objects in D_{qc}^{b}(O_{F^{r},X}).

Proof. The complex M^{•} belongs to D_{u}^{b}(O_{F^{r},X}) if and only if each ϕ_{H^{i}(M^{•})} is an isomorphism. Now (2.9.1) shows that this is the case exactly if each H^{i}(ϕ_{M^{•}}) is an isomorphism. Thus each ϕ_{H^{i}(M^{•})} is an isomorphism if and only ϕ_{M^{•}} is an isomorphism in D_{qc}^{b}(O_{F^{r},X}). □

Theorem 5.6. Suppose that f : Y → X is a morphism of smooth k-schemes.

(i) The functor f^{!} : D_{qc}^{b}(O_{F^{r},X}) → D_{qc}^{b}(O_{F^{r},Y}) restricts to a functor (which we denote by the same symbol)

f^{!} : D_{u}^{b}(O_{F^{r},X}) → D_{u}^{b}(O_{F^{r},Y}).

(ii) The functor f_{+} : D_{qc}^{+}(O_{F^{r},X}) → D_{qc}^{+}(O_{F^{r},Y}) restricts to a functor (which we denote by the same symbol)

f_{+} : D_{u}^{b}(O_{F^{r},Y}) → D_{u}^{b}(O_{F^{r},X}).

Proof. We prove part (i) first. Let M^{•} be a complex in D_{qc}^{b}(O_{F^{r},X}) whose cohomology sheaves are unit O_{F^{r},X}-modules. We must show that the cohomology sheaves of f^{!}M^{•} are unit O_{F^{r},Y} -modules.

We begin by considering the case of a single unit O_{F^{r},X}-module M. We must show that f^{!}M has unit cohomology sheaves. This can be verified locally, and so after replacing X by an open affine subset and Y be the inverse image under f of this subset we may assume that X is affine.

Let β : M → F_{X}^{r}^{∗}M be a generator of M, and let P^{•} be a resolution of M by free O_{X}-modules. (The remarks of (5.3.2) show that any unit O_{F^{r},X}-module has a generator, and it will be useful to give the proof after having made an arbitrary choice of generator.) Then F_{X}^{r∗}P^{•} is a free resolution of F_{X}^{r∗}M, and we may lift β to a morphism

β^{•} : P^{•} → F_{X}^{r∗}P^{•}.

From this we follow proposition 5.3.3 and construct a double complex

O_{F^{r},X} ⊗_{O_{X}} P^{•} ⟶^{1−β^{•}′} O_{F^{r},X} ⊗_{O_{X}} P^{•}

whose associated total complex is a free resolution of the complex

O_{F^{r},X} ⊗_{O_{X}} M ⟶^{1−β′} O_{F^{r},X} ⊗_{O_{X}} M,

which is in turn a resolution of M.

Thus f^{!}M is (up to shifting) the total complex associated to the double complex

O_{F^{r},Y} ⊗_{O_{Y}} f^{∗}P^{•} →^{1−(f^{∗}β)′} O_{F^{r},Y} ⊗_{O_{Y}} f^{∗}P^{•}.

There is a spectral sequence converging to the cohomology sheaves of this total complex, whose E_{1} terms are the horizontal cohomology groups of this double complex, which by proposition 5.3.3 are unit O_{F^{r},Y} -modules. By repeated applications of lemma 5.2 we see that the E_{∞} terms are again unit O_{F^{r},Y} -modules, and so finally we see that the same is true of the cohomology sheaves of f^{!}M.

This proves the proposition in the case of a single unit O_{F^{r},X}-module M. Now by truncation and induction on the length of a complex, combined with lemma 5.2, we deduce part (i) of the proposition for arbitrary complexes in D_{u}^{b}(X).

We turn to proving part (ii). To begin with, suppose that M is a unit O_{F^{r},Y} - module, with generator β : M → F_{Y}^{r} ^{∗}M, for some O_{Y} -module M. (As in the case of part (i), it will be useful to give the proof having made an arbitrary choice of generator.) Then M has the resolution

O_{F^{r},Y} ⊗_{O_{Y}} M →^{1−β′} O_{F^{r},Y} ⊗_{O_{Y}} M,

which by lemma 3.6.1 is acyclic for the functor O_{F^{r},X←Y} ⊗_{O_{F^{r} ,Y}} –.

Thus

f_{+}M = Rf_{∗}(f^{−1}O_{F^{r},X} ⊗_{f^{−1}O_{X}} ω_{Y/X} ⊗ (O_{F^{r},Y} ⊗_{O_{Y}} M →^{1−β′} O_{F^{r},Y} ⊗_{O_{Y}} M)).

Now using this description of f_{+}M, the cohomology sheaves of f_{+}M may be computed by a spectral sequence whose E_{1} terms are equal to (using lemma 3.6.2)

E_{1}^{i,j} = R^{j}f_{∗}(f^{−1}O_{F^{r},X} ⊗_{f^{−1}O_{X}} ω_{Y/X} ⊗_{O_{Y}} M) = O_{F^{r},X} ⊗_{O_{X}} R^{j}f_{∗}(ω_{Y/X} ⊗_{O_{Y}} M),

for i = −1 or i = 0, and vanish for all other values of i. We wish to compute the boundary map d_{1}^{−1,j}.

This boundary map is equal to H^{j}(f_{+})(1−β′) = 1−H^{j}(f_{+})β′. Now proposition 3.7.1 shows that β induces a map

γ : Rf_{∗}(ω_{Y/X} ⊗_{O_{Y}} M) → F_{X}^{r∗}Rf_{∗}(ω_{Y/X} ⊗_{O_{Y}} M),

such that f_{+}β′ is equal to the corresponding map of induced complexes

γ′ : O_{F^{r},X} ⊗_{O_{X}} Rf_{∗}(ω_{Y/X} ⊗_{O_{Y}} M) → O_{F^{r},X} ⊗_{O_{X}} Rf_{∗}(ω_{Y/X} ⊗_{O_{Y}} M).

Let us write

γ^{j} = H^{j}(γ) : R^{j}f_{∗}(ω_{Y/X} ⊗_{O_{Y}} M) → F_{X}^{r}^{∗}R^{j}f_{∗}(ω_{Y/X} ⊗_{O_{Y}} M)

(since F_{X}^{r} is flat, the pull-back F_{X}^{r}^{∗} commutes with taking cohomology) and

γ′^{j} = H^{j}(γ′) : O_{F^{r},X} ⊗_{O_{X}} R^{j}f_{∗}(ω_{Y/X} ⊗_{O_{Y}} M) → O_{F^{r},X} ⊗_{O_{X}} R^{j}f_{∗}(ω_{Y/X} ⊗_{O_{Y}} M).

Then γ′^{j} is the morphism of induced left O_{F^{r},X}-modules corresponding to the morphism γ^{j}, and we see that

d_{1}^{−1,j} = 1−γ′^{j} : O_{F^{r},X} ⊗_{O_{X}} R^{j}f_{∗}(ω_{Y/X} ⊗_{O_{Y}} M) → O_{F^{r},X} ⊗_{O_{X}} Rf^{j}(ω_{Y/X} ⊗_{O_{Y}} M).

Thus proposition 5.3.3 shows that E_{2}^{−1,j} (which is the kernel of d_{1}^{−1,j}) vanishes, and that E_{2}^{0,j} (which is the cokernel of d_{1}^{−1,j}) is the unit O_{F^{r},Y} -module generated by γ^{j}.

Thus the spectral sequence collapses to a single column at the E_{2} stage, so that H^{j}(f_{+})M = E_{2}^{0,j} is the unit O_{F^{r},Y} -module generated by γ^{j}. This concludes the proof that f_{+}M ∈ D_{u}^{b}(O_{F^{r},X}).

Now using truncation and induction on the length of a complex, together with lemma 5.2, we see that

f_{+} : D_{u}^{b}(O_{F^{r},Y}) → D_{u}^{b}(O_{F^{r},X}),

completing the proof of part (ii) of the proposition. □

(5.7) If f : Y → X is a closed immersion of smooth k-schemes then we can give an alternative proof of part (i) of the preceding lemma, which is related to the description of f^{!} provided by proposition 2.8.4. In the following discussion we let I denote the ideal sheaf cutting out Y in X.

We suppose that M is a single unit O_{F^{r},X}-module, generated by the morphism β : M → F_{X}^{r∗}M. Let J^{•} be a right resolution of M^{•} by injective O_{X}-modules. Then (since F_{X}^{r} is residually stable) we see that F_{X}^{r∗}J^{•} is an injective resolution of F_{X}^{r∗}M. The morphism β can thus be extended to a morphism

β^{•} : J^{•} → F_{X}^{r∗}J^{•}.

Let J^{•} denote the complex of unit O_{F^{r},X}-modules generated by the morphisms β^{•}. Then J^{•} is constructed as a direct limit of the modules F_{X}^{rn∗}J^{•}. Since each of these is injective as an O_{X}-module, and since X is noetherian, we see that J^{•} is a complex of unit O_{F^{r},X}-modules which are injective as O_{X}-modules. Since taking direct limits is exact, we see that J^{•} provides a right resolution of M^{•}.

Being O_{X}-injective, each J^{•} is acyclic for the functor \underline{RHom}_{O_{X}}^{•} (f_{∗}O_{Y} , –). Thus we may use the resolution J^{•} of M to compute f^{!}M, under the guise of ω_{Y/X}^{−1} ⊗_{O_{Y}} \underline{RHom}_{O_{X}}^{•} (f_{∗}O_{Y} , M), as allowed by proposition 2.4. Thus to see that f^{!}M lies in D_{u}^{b}(O_{F^{r},Y}), it suffices to show that each of the O_{F^{r},Y} -modules ω_{Y/X}^{−1} ⊗_{O_{Y}} J^{•}[I] is a unit O_{F^{r},Y} -module. This is a consequence of the following result.

Lemma 5.7.1. Suppose that M^{•} is a unit O_{F^{r},X}-module, generated by a morphism γ : M → F_{X}^{r∗}M. Then γ induces in a natural way a morphism

ω_{Y/X}^{−1} ⊗_{O_{Y}} M[I] → F_{Y}^{r∗}(ω_{Y/X}^{−1} ⊗_{O_{Y}} M[I]),

and the unit O_{F^{r},Y} -module generated by this morphism is naturally isomorphic to the O_{F^{r},Y} -module ω_{Y/X}^{−1} ⊗_{O_{Y}} M[I] constructed via corollary 2.8.2. In particular, this latter module is a unit O_{F^{r},Y} -module. (Of course, this last claim also follows from theorem 5.6, part (i), and proposition 2.8.4.)

Proof. The morphism γ induces a map

M[I^{(r)}] → (F_{X}^{r∗}M)[I^{(r)}] ⟶^{∼} F_{X}^{r′∗}(M[I]),

and consequently a morphism

M[I] → F_{X}^{r∗}(M[I])[I].

Combining this with the inverse of the isomorphism

ω_{Y/X} ⊗_{O_{Y}} F_{Y}^{r∗}(ω_{Y/X}^{−1} ⊗ M[I]) ⟶^{∼} F_{X}^{r′∗}(M[I])

constructed at the end of (A.2.2) (taking the N of that discussion to be ω_{Y/X}^{−1} ⊗ M[I]), we obtain a morphism

M[I] → ω_{Y/X} ⊗_{O_{Y}} F_{Y}^{r∗}(ω_{Y/X}^{−1} ⊗ M[I]),

and consequently a morphism

ω_{Y/X}^{−1} ⊗_{O_{Y}} M[I] → F_{Y}^{r∗}(ω_{Y/X}^{−1} ⊗ M[I]).

Tracing through the construction of corollary 2.8.2 and (hence) that of lemma 2.8.1, one verifies that this morphism does indeed generate ω_{Y/X}^{−1} ⊗_{O_{Y}} M[I]. □

(5.7.2) This recipe for passing from a generator of M to a generator of ω_{Y/X}^{−1}⊗_{O_{Y}} M[I] = H^{0}(f^{!})M is found in the proof of [Lyu, Prop. 3.1] (although it is not expressed in this language).

(5.8) The following theorem, which is the analogue for unit O_{F^{r}}-modules of Kashiwara’s theorem in the theory of D-modules, is a generalization of [Lyu, Prop. 3.1]. To see the connection between our approach and that of Lyubeznik, one should refer to the description of f^{!} provided by proposition 2.8.4 and the discussion of (5.7), as well as the description of the adjunction morphism f_{+}f^{!} → id provided by lemma 4.4.17.

Theorem 5.8.1. Let f : Y → X be a closed immersion of smooth k-schemes. If M is a unit O_{F^{r},X}-module supported on Y, then there is a natural isomorphism f_{+}f^{!}M ⟶^{∼} M. As a consequence, we see that H^{0}(f^{!})M ⟶^{∼} f^{!}M.

Proof. Since f_{+} is exact (by lemma 3.5.8), we see that if the stated isomorphism holds, then f^{!}M must be a single module in degree zero, which proves the second claim.

The isomorphism of the theorem may be verified locally, and so we may assume that X is affine and that Y is cut out by a regular sequence (a_{1}, …, a_{s}) with the property that each of the subschemes V (a_{1}, …, a_{i}) of X is smooth over k, for 1 ≤ i ≤ s. Then we may factor the closed immersion f as a product

Y = V (a_{1}, …, a_{s}) →^{f_{s}} V (a_{1}, …, a_{s−1}) →^{f_{s−1}} ⋯ →^{f_{2}} V (a_{1}) →^{f_{1}} X,

with each f_{i} being the closed immersion of a divisor. The adjunction morphism has the factorization

f_{+}f^{!}M = f_{1+} ⋯ f_{s+}f_{s}^{!} ⋯ f_{1}^{!}M

→ f_{1+} ⋯ f_{s−1 +}f_{s−1}^{!} ⋯ f_{1}^{!}M → ⋯ → f_{1+}f_{1}^{!}M → M, Since each f_{i−1}^{!} ⋯ f_{1}^{!}M is a unit O_{F^{r},V (a_{1},... ,a_{i−1})}-module which is supported on V (a_{1}, …, a_{i}), we see that to prove that

f_{+}f^{!}M → M

is an isomorphism, it also suffices to consider the case of s = 1. Thus for the remainder of the proof we assume that Y = V (a) is a smooth divisor on X, and that M is a unit O_{F^{r},X}-module supported on Y.

To prove that the adjunction map is an isomorphism, we factor it as follows:

f_{+}f^{!}M ⟶^{∼} f_{+}O_{F^{r},Y} [d_{Y/X}] ⊗^{𝕃} M ⟶^{tr_{F^{r} ,f} ⊗^{𝕃}id_{M}} O_{F^{r},X} ⊗ M ⟶^{∼} M.

(Compare the proof of theorem 4.4.1.) We will use the explicit formula of (4.4.13) to show that

(5.8.2)  f_{+}O_{F^{r},Y} [d_{Y/X}] ⊗^{𝕃} M → M

is an isomorphism.

Since M is supported on Y, every section of M is annihilated by some power of a, and so M = ⋃_{n=0}^{∞} M[a^{q^{n}}]. Base-change by the flat morphism F_{X}^{rn∗} shows that

F_{X}^{rn∗}(M[a]) ⟶^{∼} (F_{X}^{rn∗}M)[a^{q^{n}}].

Composing with ϕ_{M} we find that F_{X}^{rn∗}(M[a]) ⟶^{∼} M[a^{q^{n}}]. Thus M is generated by the inclusion

M[a] → M[a^{q}] ⟶^{∼} F_{X}^{r∗}(M[a]),

and we have the consequent presentation

0 → O_{F^{r},X} ⊗_{O_{X}} M[a] → O_{F^{r},X} ⊗_{O_{X}} M[a] → M → 0

of M.

We now compute (5.8.2) using (4.4.13) and this presentation. The source of (5.8.2) is represented by the tensor product

(O_{F^{r},X} → O′_{F^{r},X}) ⊗_{O_{F^{r} ,X}} (O_{F^{r},X} ⊗_{O_{X}} M[a] → O_{F^{r},X} ⊗_{O_{X}} M[a]).

(Here we are using the observation of (4.4.13) that the modules O_{F^{r},X} and O′_{F^{r},X} are acyclic with respect to tensor product by induced left O_{F^{r},X}-modules.) To prove that (5.8.2) is an isomorphism, it suffices to show that

O′_{F^{r},X} ⊗_{O_{F^{r} ,X}} (O_{F^{r},X} ⊗_{O_{X}} M[a] → O_{F^{r},X} ⊗_{O_{X}} M[a])

⟶^{∼} (O′_{F^{r},X} ⊗_{O_{X}} M[a] → O′_{F^{r},X} ⊗_{O_{X}} M[a])

is an acyclic complex. Let d denote the differential of this complex.

Since O_{F^{r},X′} is isomorphic to O_{F^{r},X} as a right O_{X}-module, we see (by (1.3.2)) that

O_{F^{r},X′} ⊗_{O_{X}} M[a] ⟶^{∼} O_{F^{r},X} ⊗_{O_{X}} M ⟶^{∼} ⨁_{n=0}^{∞} M[a^{q^{n}}],

and so the morphism d is determined by its restriction to M[a]. Using the explicit description of the right O_{F^{r},X}-module structure of O′_{F^{r},X}, we see that this restriction is given by

M[a] ⟶^{m↦m⊕−a^{q−1}m} M[a] ⊕ M[a^{q}].

Since a^{q−1} annihilates M[a], this map is simply the identity map from M[a] to itself. This shows that d is an isomorphism, and completes the proof of the theorem. □

We have the following corollary of this result:

Corollary 5.8.3. If f : Y → X is a closed immersion of smooth k-schemes, and M is a unit O_{F^{r},X}-module, than the adjunction morphism

f_{+}H^{0}(f^{!})M → M

is an injection, and identifies f_{+}H^{0}(f^{!})M with the subsheaf Γ_{Y} (M) of M consisting of sections of M supported on Y.

Proof. Since Γ_{Y} (M) = ⋃_{n=1}^{∞} M[I^{n}], and F^{r}(M[I^{n}]) ⊂ M[I^{qn}], one sees that Γ_{Y} (M) is a sheaf of left O_{F^{r},X}-submodules of M. Furthermore, the structural isomorphism ϕ_{M} induces an isomorphism F_{X}^{r}^{∗}(Γ_{Y} (M)) ⟶^{∼} Γ_{Y} (F_{X}^{r}^{∗}(M)) →^{∼} Γ_{Y} (M). Thus Γ_{Y} (M) is a unit O_{F^{r},X}-module.

Now Γ_{Y} (M)[I] = M[I], and so one concludes by proposition 2.8.4 that the natural morphism H^{0}(f^{!})Γ_{Y} (M) → H^{0}(f^{!})M is an isomorphism. Thus

f_{+}H^{0}(f^{!})M ⟶^{∼} f_{+}H^{0}(f^{!})Γ_{Y} (M) ⟶^{∼} Γ_{Y} (M),

where the second isomorphism is given by theorem 5.8.1. □

(5.9) We will also deduce an analogue of theorem 5.8.1 for the derived category. First, some definitions:

Definition 5.9.1. If X is a k-scheme, and if M^{•} is an object of D^{b}(O_{F^{r},X}), the support of M^{•} is defined to be the union of the support of the cohomology sheaves H^{i}(M^{•}), for every i. (Note that since H^{i}(M^{•}) = 0 for all but finitely many i, this reduces to a finite union, and so the support of M^{•} is a closed subset of X.)

Definition 5.9.2. If Z is a closed subset of a smooth k-scheme X, we define D_{u,Z}^{b}(O_{F^{r},X}) to be the full triangulated subcategory of D_{u}^{b}(O_{F^{r},X}) consisting of objects whose support is contained in Z.

Corollary 5.9.3. Let f : Y → X be a closed immersion of smooth k-schemes. Then the essential image of the functor

f_{+} : D_{u}^{b}(O_{F^{r},Y}) → D_{u}^{b}(O_{F^{r},X})

is equal to D_{u,Y}^{b} (O_{F^{r},X}).

Furthermore, f_{+} induces an equivalence of categories between D_{u}^{b}(O_{F^{r},Y}) and D_{u,Y}^{b} (O_{F^{r},X}), with f^{!} providing a quasi-inverse.

Proof. This follows from theorem 5.8.1 by truncation and induction on the length of a complex in D_{u,Y}^{b} (O_{F^{r},X}). □

(5.9.4) Recall that if Y is a closed subset of X then the functor RΓ_{Y} is defined as the total right derived functor of the functor Γ_{Y} of sections with support in Y. If M^{•} is a bounded complex of left O_{F^{r},X}-modules, then RΓ_{Y} (M^{•}) is naturally a complex of left O_{F^{r},X}-modules, since we may compute RΓ_{Y} by taking a resolution of M^{•} by injective left O_{F^{r},X}-modules (1.3.1). By a result of Grothendieck, RΓ_{Y} has finite cohomological amplitude. Thus RΓ_{Y} (M^{•}) has only finitely many nonvanishing cohomology sheaves and so is an object of D^{b}(O_{F^{r},X}).

The following result generalizes [Lyu, Example 1.2 (b)], and provides an analogue of corollary 5.8.2 for the derived category. It provides a strengthening of lemma 4.5.3 for unit O_{F^{r},X}-modules.

Proposition 5.9.5. If f : Y → X is a closed immersion of a k-scheme Y into a smooth k-scheme X and M^{•} is an object of D_{u}^{b}(O_{F^{r},X}), then RΓ_{Y} (M^{•}) is an object of D_{u,Y}^{b} (O_{F^{r},X}). If furthermore Y is smooth k-scheme, then there is a natural isomorphism

RΓ_{Y} (M^{•}) ⟶^{∼} f_{+}f^{!}M^{•}

of objects in D_{u,Y}^{b} (O_{F^{r},X}).

Proof. Let j : X \ Y → X be the open immersion of the complement of Y. Then we have the distinguished triangle of objects of D^{b}(O_{F^{r},X}))

⋯ → RΓ_{Y} (M^{•}) → M^{•} → j_{+}j^{!}M^{•} → · · · .

Since M^{•} and j_{+}j^{!}M^{•} are both objects of D_{u}^{b}(O_{F^{r},X}) we see that the same is true of RΓ_{Y} (M^{•}). By construction, it has support in Y.

If furthermore Y is smooth as a k-scheme then we may apply f^{!} to the above distinguished triangle to obtain the distinguished triangle

⋯ → f^{!}RΓ_{Y} (M^{•}) → f^{!}M^{•} → 0 → ⋯

of objects in D_{u}^{b}(O_{F^{r},Y}). Thus f^{!}RΓ_{Y} (M^{•}) ⟶^{∼} f^{!}M^{•}. As RΓ_{Y} (M^{•}) is an object of D_{u,Y}^{b} (O_{F^{r},X}), corollary 5.9.3 gives

RΓ_{Y} (M) ⟶^{∼} f_{+}f^{!}RΓ_{Y} (M^{•}) ⟶^{∼} f_{+}f^{!}M^{•}.

□

Example 5.9.6 Consider the case when M^{•} = O_{X}. Then f^{!}O_{X} = O_{Y} [d_{Y/X}]. Thus we see that in the situation of the preceding proposition

f_{+}O_{Y} ⟶^{∼} RΓ_{Y} (O_{X})[d_{X/Y}].

(5.10) We present one final application of theorem 5.8.1.

Proposition 5.10.1. Let X be a smooth k-scheme, and let M^{•} be an object of D_{u}^{b}(O_{F^{r},X}) with support contained in the closed subset Z of X. Let U be any open subscheme of Z which (given its reduced induced structure) is smooth as a k-scheme, and let f : U → X denote the immersion of U into X. Then there is a natural morphism

M^{•} → f_{+}f^{!}M^{•},

whose cone is supported on Z \ U.

Proof. Let V = X \ (Z \ U). Then U is closed in V and V is open in X. Let i : U → V and j : V → X denote the corresponding immersions, so that f = ji. Then by lemma 4.3.1 there is the natural morphism of adjunction

(5.10.2)  M^{•} → j_{+}j^{!}M^{•}.

Now since M^{•} is supported on Z, we see that j^{!}M^{•} is supported on Z ∩ V = U. Thus by corollary 5.9.3 we see that

(5.10.3)  j^{!}M^{•} ⟶^{∼} i_{+}i^{!}j^{!}M^{•}.

This induces an isomorphism

j_{+}j^{!}M^{•} →^{∼} j_{+}i_{+}i^{!}j^{!}M^{•} = f_{+}f^{!}M^{•}.

Composing with the morphism (5.10.2) yields a morphism

ψ : M^{•} → j_{+}i_{+}i^{!}j^{!}M^{•} = f_{+}f^{!}M^{•}.

Applying j^{!} to ψ recovers the isomorphism (5.10.3), and so the cone of ψ is supported on X \ V = Z \ U. This proves the proposition. □

(5.11) Let X be a smooth k-scheme. If k′ is an extension field of k, and X′ is the base-change of X over k′, then base-change of O_{F^{r},X}-modules via k′ induces functors

k′ ⊗_{k} – : μ_{u}(X) → μ_{u}(X′)

and

k′ ⊗_{k} – : D_{u}^{b}(O_{F^{r},X}) → D_{u}^{b}(O_{F^{r},X′}).

A particular case of interest is that in which k′ is a purely inseparable algebraic extension of k.

Proposition 5.11.1. Let k′ be a (necessarily purely inseparable algebraic) extension of k such (k′)^{q^{n}} is contained in k for some n, let X be a smooth k-scheme, and let X′ = k′ ×_{k} X be the base-change of X over k′.

(i) For any two complexes M^{•} and N^{•} in D_{u}^{b}(O_{F^{r},X}), the natural morphism

\underline{RHom}_{O_{F^{r} ,X}}^{•} (M^{•}, N^{•}) → \underline{RHom}_{O_{F^{r} ,X′}}^{•} (k′ ⊗_{k} M^{•}, k′ ⊗_{k} N^{•})

of (1.13.5) is an isomorphism.

(ii) The functor k′ ⊗_{k} – : μ_{u}(X) → μ_{u}(X′) induces an equivalence of categories. As a consequence of (i) and (ii) we deduce that

k′ ⊗_{k} – : D_{u}^{b}(O_{F^{r},X}) → D_{u}^{b}(O_{F^{r},X′})

induces an equivalence of categories.

Proof. By considering successive changes of ground field and induction on n, we may reduce to the case that (k′)^{q} ⊂ k.

Since M^{•} and N^{•} are unit O_{F^{r},X}-modules, we deduce from the commutative diagram of (2.9.2) that the morphism

\underline{RHom}_{O_{F^{r} ,X}}^{•} (M^{•}, N^{•}) → \underline{RHom}_{O_{F^{r} ,X}}^{•} (F_{X}^{r∗}M^{•}, F_{X}^{r∗}N^{•})

is an isomorphism in D^{+}(X), and similarly for the morphism

\underline{RHom}_{O_{F^{r} ,X′}}^{•} (k′ ⊗_{k} M^{•}, k′ ⊗_{k} N^{•}) → \underline{RHom}_{O_{F^{r} ,X′}}^{•} (F_{X′}^{r∗}(k′ ⊗_{k} M^{•}), F_{X′}^{r∗}(k′ ⊗_{k} N^{•})).

Now in the commutative diagram of lemma 2.9.4 we see that the two horizontal arrows are isomorphisms. This is sufficient to conclude that the left-hand vertical arrow is an isomorphism, proving (i).

Since (i) shows that the functor of (ii) is fully faithful, to establish (ii) it remains to show that any object M′ of μ_{u}(X′) is isomorphic to k′ ⊗_{k} M for some object M of μ_{u}(X). As in the proof of (i), we may assume that (k′)^{q} ⊂ k. Via this inclusion, we obtain a diagram

\xymatrix{X′ \ar[r] \ar[d] & X \ar[r] \ar[d] & X′ \ar[d] \\ Spec k′ \ar[r] & Spec k \ar[r] & Spec k′}

such that the composition of the top two horizontal arrows is F_{X′}^{r} and the composition of the bottom two horizontal arrows is F_{k′}^{r}. If we define M = k ⊗_{k′} M′ (where k is regarded as a k′-algebra via the inclusion (k′)^{q} ⊂ k) then M is an object of μ_{u}(X) and k′ ⊗_{k} M = F_{X′}^{r}^{∗}M′. But the structural morphism ϕ_{M′} provides an isomorphism between the O_{F^{r},X′}-module F_{X′}^{r}^{∗}M′ and M′. Thus M′ is isomorphic to k′ ⊗_{k} M. □

Proposition 5.12. If X is a smooth k-scheme and M is a unit O_{F^{r},X}-module then \underline{RHom}_{O_{F^{r} ,X}}^{•} (M, –) has finite cohomological dimension. Consequently, if M^{•} is a complex lying in D_{u}^{b}(O_{F^{r},X}) and N^{•} is a complex lying in D^{b}(O_{F^{r},X}) then \underline{RHom}_{O_{F^{r} ,X}}^{•} (M^{•}, N^{•}) is a bounded complex of 𝔽_{q}-sheaves.

Proof. Let M and N be unit O_{F^{r},X}-modules. We must show that \underline{Ext}_{O_{F^{r} ,X}}^{i} (M, N) vanishes for sufficiently large i. This may be verified by working locally on X, and so we may assume that X is affine. If β : M → F_{X}^{r∗}M generates M, then as in the proof of part (i) of theorem 5.6, we choose a free resolution P^{•} of M and a lift of β to β^{•} : P^{•} → F_{X}^{r∗}P^{•}. Now since X is smooth over k, and so regular, we may choose P to be finite length. The total complex of the double complex

O_{F^{r},X} ⊗_{O_{X}} P^{•} ⟶^{1−β^{•′}} O_{F^{r},X} ⊗_{O_{X}} P^{•}

then yields a finite length free resolution of M. We may use this to compute \underline{RHom}_{O_{F^{r} ,X}}^{•} (M, N), and the proposition follows. □

6. locally finitely generated unit O_{F^{r}}-modules

(6.1) In [Lyu] there is defined a notion of F -finite module, which in our terminology is a unit O_{F,X}-module over an affine scheme X which can be generated by a morphism β : M → F^{∗}M with M a coherent O_{X}-module. Our first aim in this section is to show that a unit O_{F^{r},X}-module admits such a generator if and only if M is locally finitely generated as an O_{F^{r},X}-module. We first state a definition, which is an adaptation of a notion of [Lyu].

Definition 6.1.1. A root of a unit O_{F^{r},X}-module M on a k-scheme X is an injective morphism β : M → F_{X}^{r∗}M, with M a coherent O_{X}-module, which generates M.

Remark 6.1.2 By definition, if a unit O_{F^{r},X}-module has a root, than it has a coherent generator. Since coherent O_{X}-modules are locally finitely generated as O_{X}-modules, we see from Proposition 5.3.3 that any unit O_{F^{r},X}-module which has a coherent generator is locally finitely generated as an O_{F^{r},X}-module. The following result completes this chain of implications, by showing that all three conditions are equivalent:

Theorem 6.1.3. Let X be a smooth k-scheme. If M is a unit O_{F^{r},X}-module which is locally finitely generated (as a left O_{F^{r},X}-module), then M has a root.

Proof. The quasi-coherent module M is equal to the direct limit of its O_{X}-coherent sub-modules. Since M is locally finitely generated as an O_{F^{r},X}-module, there exists an O_{X}-coherent submodule M ⊂ M which generates M. The morphism O_{F^{r},X} ⊗_{O_{X}} M → M is then surjective.

Pulling back by F_{X}^{r∗}, we see that F_{X}^{r}^{∗}(O_{F^{r},X} ⊗_{O_{X}} M) → F_{X}^{r}^{∗}M is surjective. By corollary 1.3.2,

F_{X}^{r}^{∗}(O_{F^{r},X} ⊗_{O_{X}} M) = F_{X}^{r}^{∗}(⨁_{n=0}^{∞}(F_{X}^{rn})^{∗}M) = ⨁_{n=1}^{∞}(F_{X}^{rn})^{∗}M.

Thus we have a surjection ⨁_{n=1}^{∞}(F_{X}^{rn})^{∗}M → F_{X}^{r∗}M.

Consider the isomorphism ϕ_{M}^{−1} : M → F_{X}^{r∗}M. Since M is coherent, its image under ϕ_{M}^{−1} is contained in the sum of the images of (F_{X}^{rn})^{∗}M for finitely many values of n, say 1 ≤ n ≤ N.

Let M′ = ∑_{n=0}^{N−1} ϕ_{n,M}F_{X}^{rn∗}M ⊂ M. Then we see that

ϕ_{M}(F_{X}^{r∗}M′) = ∑_{n=1}^{N} ϕ_{n,M}(F_{X}^{rn∗}M) ⊂ M.

Define β to be the restriction of ϕ_{M}^{−1} to M′. Then we see from this that β : M′ → F_{X}^{r∗}M′. This morphism generates M by construction, and is injective with coherent domain, again by construction. This proves the theorem. □

(6.2) The following results generalize some of the more elementary results of [Lyu, §2] (once one makes the translation from locally finitely generated unit O_{F} - modules to modules admitting a coherent generator).

Lemma 6.2.1. If

0 → M → M → M′′ → 0

is a short exact sequence of unit O_{F^{r},X}-modules on a smooth k-scheme X, then M is locally finitely generated if and only if both M′ and M′′ are locally finitely generated.

Proof. The proof of this result follows that of the proof of [Lyu, Theorem 2.8]. The key ingredient in this argument is the construction of a root of a locally finitely generated unit O_{F^{r},X}-module M, and this is provided by theorem 6.1.3. □

Corollary 6.2.2. If

M_{1} → M_{2} → M_{3} → M_{4} → M_{5}

is an exact sequence of O_{F^{r},X}-modules and M_{1}, M_{2}, M_{4} and M_{5} are locally finitely generated unit, then M_{3} is locally finitely generated unit.

Proof. It follows from lemma 5.2 that M_{3} is a unit O_{F^{r},X}-module. Let M′ be the cokernel of the morphism M_{1} → M_{2}. Then by lemma 5.2, M′ is a unit O_{F^{r},X}- module, and so by the preceding lemma, M′ is locally finitely generated. A similar argument shows that the kernel M′′ of M_{4} → M_{5} is locally finitely generated. Since M_{3} is an extension of M′ by M′′, we see by the preceding lemma that M_{3} is locally finitely generated. □

Lemma 6.3. If M and N are locally finitely generated unit O_{F^{r},X}-modules, then product M ⊗_{O_{X}} N is a locally finitely generated unit O_{F^{r},X}-module.

Proof. This follows immediately from theorem 6.1.3, lemma 5.4.1 and the fact that the tensor product of coherent O_{X}-modules is a coherent O_{X}-module. □

Definition 6.4. We let μ_{lfgu}(X) denote the full subcategory of μ(X) consisting of locally finitely generated unit O_{F^{r},X}-modules. We let D_{lfgu}^{b}(O_{F^{r},X}) denote the full triangulated subcategory of D_{qc}^{b}(O_{F^{r},X}) consisting of complexes whose cohomology sheaves are locally finitely generated unit O_{F^{r},X}-modules.

It follows by the preceding corollary that μ_{lfgu}(X) is a thick subcategory of μ(X), so that the full subcategory D_{lfgu}^{b}(O_{F^{r},X}) of D^{b}(O_{F^{r},X}) is indeed triangulated. This is the category for which we will eventually prove our Riemann-Hilbert correspondence.

Proposition 6.5. If f : Y → X is a morphism of smooth k-schemes, then the functor f^{!} : D_{qc}^{b}(O_{F^{r},X}) → D_{qc}^{b}(O_{F^{r},X}) restricts to a functor (which we denote by the same symbol)

f^{!} : D_{lfgu}^{b}(O_{F^{r},X}) → D_{lfgu}^{b}(O_{F^{r},X}).

Proof. Let M^{•} be a complex in D_{qc}^{b}(O_{F^{r},X}) whose cohomology sheaves are locally finitely generated unit O_{F^{r},X}-modules. We must show that the same is true of the cohomology sheaves of f^{!}M^{•}.

If we consider the proof of part (i) of theorem 5.6, and note that corollary 6.2.2 provides the analogue in the context of locally finitely generated unit O_{F^{r},X}- modules of lemma 5.2, we see that it is enough to prove the proposition in the case of a single locally finitely generated unit O_{F^{r},X}-module M. We must show that f^{!}M has locally finitely generated cohomology sheaves.

Since this can be verified locally, after replacing X by an open affine subset and Y by the inverse image under f of this subset, we may assume that X is affine.

By theorem 6.1.3 we may choose a generator β : M → F_{X}^{r}^{∗}M of M with M a coherent O_{X}-module. Following the proof of part (i) of theorem 5.6 , we choose a free resolution P^{•} of M. Since M is coherent, we may choose each P^{j} to be free of finite rank. We then choose a lift of β to a morphism β^{•} : P^{•} → F_{X}^{r∗}P^{•}, and

construct the double complex O_{F^{r},X} ⊗_{O_{X}} P^{•} →^{1−β^{•}′} O_{F^{r},X} ⊗_{O_{X}} P^{•}, which as in the proof of 5.6(i) gives rise to a spectral sequence computing the cohomology sheaves of f^{!}M. The E_{1} terms of the spectral sequence are the horizontal cohomology groups of this double complex, which are locally finitely generated unit O_{F^{r},Y} -modules, since the P^{j} are of finite rank. By repeated applications of corollary 6.2.2 we see that the E_{∞} terms are again locally finitely generated unit O_{F^{r},Y} -modules, and so finally the cohomology sheaves of f^{!}M are locally finitely generated unit O_{F^{r},Y} - modules. This completes the proof of the proposition. □

(6.6) The analogous result for f_{+} is also true, but the proof is more involved.

Proposition 6.6.1. Let f : Y → X be an open immersion of smooth k-schemes such that the complement of the image of f is a divisor on X. If M is a locally finitely generated unit O_{F^{r},Y} -module M, then H^{0}(f_{+})M = f_{+}M is a locally finitely generated unit O_{F^{r},X}-module.

Proof. By part (ii) of theorem 5.6 we see that f_{+}M is an object of D_{u}^{b}(O_{F^{r},X}). However, since f is the open immersion of the complement of a divisor, the higher derived direct images of f vanish when applied to quasi-coherent O_{Y} -modules, and so by lemma 4.3.1

f_{+}M = Rf_{∗}M = H^{0}(Rf_{∗}M) = f_{∗}M

is in fact an object of μ(X).

Let D = X \ Y be the complement of Y, and let O_{X}(D) denote the associated invertible sheaf. Let

β : M → F_{Y}^{r} ^{∗}M

be a generator of M, with M a coherent O_{Y} -module. Then applying f_{∗} we obtain a map

f_{∗}(β) : f_{∗}M → f_{∗}F_{Y}^{r} ^{∗}M ⟶^{∼} F_{X}^{r}^{∗}f_{∗}M.

This morphism generates f_{∗}M, but f_{∗}M is not coherent. To find a coherent generator, we argue as in the proof of [Lyu, Prop. 2.9(b)].

Let N be a coherent submodule of f_{∗}M such that f^{∗}N = M. Then

f_{∗}M = ⋃_{n=0}^{∞} N ⊗_{O_{X}} O_{X}(D)^{⊗n},

while

F_{X}^{r}^{∗}(f_{∗}M) = ⋃_{n=0}^{∞} F_{X}^{r}^{∗}N ⊗_{O_{X}} O_{X}(D)^{⊗n}.

Since N is coherent, there is an integer n of the form n = (q −1)m for some positive integer m such that f_{∗}(β)(N) ⊂ F_{X}^{r}^{∗}N ⊗_{O_{X}} O_{X}(D)^{⊗n}. Let us denote by γ′ the morphism γ′ : N → F_{X}^{r}^{∗}N ⊗_{O_{X}} O_{X}(D)^{⊗n} obtained by restricting f_{∗}(β) to N and by γ the morphism

γ = γ′ ⊗ id_{O_{X}(D)^{⊗m}} : N ⊗_{O_{X}} O_{X}(D)^{⊗m}

→ F_{X}^{r}^{∗}N ⊗_{O_{X}} O_{X}(D)^{⊗qm} = F_{X}^{r}^{∗}(N ⊗_{O_{X}} O_{X}(D)^{⊗m}).

Then γ is the generating morphism for a unit O_{F^{r},X}-module, which is immediately seen to be f_{∗}M, since m was chosen to be positive. Since N ⊗_{O_{X}} O_{X}(D)^{⊗m} is coherent we see that f_{∗}M is locally finitely generated, proving the proposition. □

Proposition 6.6.2. If f : Y → X is a proper morphism of smooth k-schemes, then the functor f_{+} : D_{qc}^{b}(O_{F^{r},Y}) → D_{qc}^{b}(O_{F^{r},X}) restricts to a functor (which we denote by the same symbol)

f_{+} : D_{lfgu}^{b}(O_{F^{r},Y}) → D_{lfgu}^{b}(O_{F^{r},X}).

Proof. The proof of this follows the proof of part (ii) of theorem 5.6. We note that lemma 6.2.2 provides the analogue in the case of locally finitely generated unit modules of lemma 5.2. As in that proof, we may reduce to the case of a single locally finitely generated unit module M, generated by β : M → F_{Y}^{r} ^{∗}M, with M a coherent O_{Y} -module.

Then, in the notation of that proof, we see that H^{j}f_{+}M is the unit module generated by

γ^{j} : R^{j}f_{∗}(ω_{Y/X} ⊗_{O_{Y}} M) → F_{X}^{r}^{∗}R^{j}f_{∗}(ω_{Y/X} ⊗_{O_{Y}} M).

Since f is proper, and ω_{Y/X} ⊗_{O_{Y}} M is coherent, we see that R^{j}f_{∗}(ω_{Y/X} ⊗_{O_{Y}} M) is coherent. Thus indeed H^{j}f_{+}M is a locally finitely generated unit O_{F^{r},Y} -module, and the proposition is proved. □

Corollary 6.6.3. If f : Y → X is an immersion of smooth k-schemes, then the functor f_{+} : D_{qc}^{+}(O_{F^{r},Y}) → D_{qc}^{b}(O_{F^{r},X}) restricts to a functor (which we denote by the same symbol)

f_{+} : D_{lfgu}^{b}(O_{F^{r},Y}) → D_{lfgu}^{b}(O_{F^{r},X}).

Proof. We factor f = f_{1}f_{2}, with f_{1} an open immersion and f_{2} a closed immersion. To prove the corollary, it suffices to prove it for open and closed immersions separately.

Suppose first that f is an open immersion, and that M is a locally finitely generated unit O_{F^{r},Y} -module on X. We have to show that f_{+}M has locally finitely generated unit cohomology sheaves.

We can check this by restricting f_{+}M to affine open subschemes of X, and so we may assume that X = Spec A is affine. Let Y be the complement of the closed subset V (a_{1}, …, a_{s}) cut out by the elements a_{1}, …, a_{s} of A.

By lemma 4.3.1 we see that f_{+}M = Rf_{∗}M. We may compute Rf_{∗}M by using the Čech complex associated to the covering {X_{a_{i}}}_{1≤i≤s} of Y. Then Rf_{∗}M is represented by a complex which in each degree is a direct sum of terms of the form j_{∗}M|_{U}, where U is the complement of a divisor in X, and j : U → X is the natural inclusion. By proposition 6.6.1, each term in this complex is a locally finitely generated unit O_{F^{r},X} module, hence its cohomology sheaves have the same property.

Now by truncation and induction on the length of a complex, together with lemma 6.2.2, we see that for any complex M^{•} in D_{lfgu}^{b}(Y), its push-forward f_{+}M is a complex in D_{lfgu}^{b}(X). This completes the proof of the proposition for an open immersion.

Suppose that f is a closed immersion. Then in particular f is proper, and so this case follows from proposition 6.7.2. □

Corollary 6.6.4. If f : Y → X is a morphism of smooth k-schemes, then the functor f_{+} : D_{qc}^{+}(O_{F^{r},Y}) → D_{qc}^{+}(O_{F^{r},X}) restricts to a functor (which we denote by the same symbol)

f_{+} : D_{lfgu}^{b}(O_{F^{r},Y}) → D_{lfgu}^{b}(O_{F^{r},X}).

Proof. Suppose first that Y is quasi-projective, and choose an immersion g : Y → ℙ_{k}^{n}, for some positive integer n. Then we may factor f as

Y ⟶^{Γ_{f}} X × Y ⟶^{id_{X}×g} X × ℙ_{k}^{n} ⟶^{p_{1}} X,

where Γ_{f} is the graph of f and p_{1} is the projection onto the first factor.

It suffices to prove the corollary for each of the morphisms of this factorization. Since Γ_{f} and id_{X}×g are both immersions, the result follows for these two morphisms by corollary 6.6.3. The result follows for the proper morphism p_{1} by theorem 6.6.2. This completes the proof of the corollary if Y is quasi-projective.

For the general case, let M^{•} be in D_{lfgu}^{b}(O_{F^{r},Y}). By theorem 5.6 (ii) we know that f_{+}M^{•} is a bounded complex of unit O_{F^{r},X}-modules. To show that f_{+}M^{•} is in D_{lfgu}^{b}(O_{F^{r},X}) we proceed by induction on the support Z of M^{•}. Choose a dense open affine U ⊂ Y such that U ∩ Z is dense in Z, and denote by j : U → Y the natural inclusion. As the cone of the adjunction morphism M^{•} → j_{+}j^{−1}M^{•} is supported on Z \ (Z ∩ U), it is enough to prove the result for j_{+}j^{−1}M^{•}, by 6.2.2. Using 3.8, we see that we may replace Y by U, and M by j^{−1}M. As U is affine, the result now follows from the quasi-projective case. □

(6.7) One particular class of locally finitely generated unit O_{F^{r},X}-modules consists of those unit O_{F^{r},X}-modules which are coherent as O_{X}-modules. For if M is such a unit O_{F^{r},X}-module then the inverse of the structural morphism ϕ_{M}^{−1} : M → F_{X}^{r}^{∗}M is a generator of M whose domain is coherent.

Lemma 6.7.1. If M is a unit O_{F^{r},X}-module on X which is coherent as an O_{X}- module, then M is locally free of finite rank as an O_{X}-module.

Proof. Suppose that M is a unit O_{F^{r},X}-module which is coherent as an O_{X}-module. It suffices to show that the stalk of M at x is free for any point x of X.

Let O_{X,x} be the stalk of O_{X} at x, a local ring with maximal ideal 𝔪_{x} and residue field κ(x). Choose a surjection

(6.7.2)  O_{X,x}^{n} → M_{x}

with n minimal (so that n equals the rank of the κ(x)-vector space M_{x}/𝔪_{x}); then this surjection becomes an isomorphism after reducing modulo 𝔪_{x}. Thus if we let R denote the kernel of (6.7.2) then R ⊂ 𝔪_{x}O_{X,x}^{n}.

Applying F_{X}^{r∗} to (6.7.2) yields a surjection

(6.7.3)  O_{X,x}^{n} = F_{X}^{r∗}O_{X,x}^{n} → F_{X}^{r∗}M_{x}

whose kernel is O_{X,x}R^{q} (that is, the module generated by the vectors obtained by raising the elements of R to the q^{th} power coordinate-wise).

We may find a morphism A : O_{X,x}^{n} → O_{X,x}^{n} (which is simply an n × n matrix with coefficients in O_{X,x}) which makes the following diagram commute:

\xymatrix{O_{X,x}^{n} \ar[r]^{(6.7.3)} \ar[d]^{A} & F_{X}^{r∗}M_{x} \ar[d]^{ϕ_{M}} \\ O_{X,x}^{n} \ar[r]^{(6.7.2)} & M_{x}.}

Upon reducing this diagram modulo 𝔪_{x}, the two horizontal arrows become isomorphisms. Since the right vertical arrow is also an isomorphism we find that A becomes an isomorphism when reduced modulo 𝔪_{x}. Thus the determinant of A is a unit modulo 𝔪_{x}, and so is a unit in the local ring O_{X,x}, showing that A is an isomorphism.

We conclude that R = AO_{X,x}R^{q}. Proceeding by induction we see that

R = A^{1+q+···+q^{m}}O_{X,x}R^{q^{m+1}} ⊂ 𝔪_{x}^{q^{m+1}}O_{X,x}^{n}

for all integers m. This implies that R = 0, hence that (6.7.2) is an isomorphism, and so we conclude that M_{x} is indeed a free O_{X,x}-module. □

Definition 6.7.4. If M is a unit O_{F^{r},X}-module which is coherent as an O_{X}- module (and hence locally free of finite rank, by lemma 6.7.1), we say that M is a unit F^{r}-crystal on X.

In order to see that this terminology agrees with the general usage of the term “F^{r}- crystal”, we should explain why such a module M is canonically endowed with an integrable connection. However, we postpone this discussion until part II [EK].

Proposition 6.7.5. Assume that k is a perfect field of characteristic p, let X be a smooth k-scheme, and let M be a locally finitely generated unit O_{F^{r},X}-module on X with support contained in the closed subset Z of X. Then there is a dense open subscheme U ⊂ Z (given its reduced induced structure) which is smooth as a k-scheme, such that, if f : U → X denotes the immersion of U in X, then f^{!}M is a unit F^{r}-crystal on U.

Proof. Since k is perfect, we may find a dense open subscheme V of Z (given its induced reduced structure) which is smooth as a k-scheme. Let W = X \ (Z \ V). Then W is an open subscheme of X and V a closed subset of W. Let g be the open immersion of W in X, and i the closed immersion of V in W. Then g^{!}M is the simply the sheaf-theoretic restriction g^{−1}M of M to W, which by assumption is supported on V. Thus by theorem 5.8.1 together with proposition 6.5 we see that i^{!}g^{!}M = H^{0}(i^{!}g^{!}M) is a locally finitely generated unit O_{F^{r},V} -module, which thus admits a root, by theorem 6.1.3. That is, we may find a generator β : N → F_{V}^{r∗}N with N coherent on V and β injective. Then we may find a dense open subscheme U of V on which N is locally free of finite rank and β is an isomorphism. If j is the open immersion of U in V then we see that j^{!}i^{!}g^{!}M = (gij)^{!}M is a unit crystal on U. □

Lemma 6.7.6. If f : Y → X is a morphism of smooth k-schemes and M is a unit F^{r}-crystal on X, then f^{!}M[d_{X/Y}] is a unit F^{r}-crystal on Y.

Proof. By lemma 6.7.1 we see that M is locally free of finite rank as an O_{X}-module. Thus

f^{!}M[d_{X/Y}] ⟶^{∼} O_{Y} ⊗^{𝕃}_{f^{−1}O_{X}} f^{−1}M ⟶^{∼} O_{Y} ⊗_{f^{−1}O_{X}} f^{−1}M

is locally free of finite rank as an O_{Y} -module. This proves the lemma. □

Proposition 6.8. Let X be a smooth k-scheme. Then the functor

– ⊗^{𝕃}_{O_{X}} – : D_{qc}^{b}(O_{F^{r},X}) × D_{qc}^{b}(O_{F^{r},X}) → D_{qc}^{b}(O_{F^{r},X})

restricts to a functor

– ⊗^{𝕃}_{O_{X}} – : D_{lfgu}^{b}(O_{F^{r},X}) × D_{lfgu}^{b}(O_{F^{r},X}) → D_{lfgu}^{b}(O_{F^{r},X}).

Proof. Given two objects M^{•} and N^{•} of D_{lfgu}^{b}(O_{F^{r},X}), we must show that the

object M^{•} ⊗^{𝕃}_{O_{X}} N^{•} of D_{qc}^{b}(O_{F^{r},X}) in fact lies in D_{lfgu}^{b}(O_{F^{r},X}). Using inseparable descent, we may replace k by its perfect closure, and so we may and do assume that k is perfect.

We begin by supposing that M^{•} is a unit F^{r}-crystal M placed in degree zero. In this case, since M is locally free as an O_{X}-module, we see that

H^{i}(M ⊗^{𝕃}_{O_{X}} N^{•}) ⟶^{∼} H^{i}(M ⊗_{O_{X}} N^{•}) ⟶^{∼} M ⊗_{O_{X}} H^{i}(N^{•}).

Now each H^{i}(N^{•}) is a locally finitely generated unit O_{F^{r},X}-module, and so lemma 6.3 shows that M ⊗_{O_{X}} H^{i}(N^{•}) is a locally finitely generated unit O_{F^{r},X}-module. This proves the proposition in this case.

We now prove the proposition for an arbitrary complex M^{•}, arguing by induction on the dimension of its support. Thus let Z be the support of M^{•}, and assume that the proposition is known for all complexes with support of dimension less than that of Z.

By truncation and induction on the length of M^{•}, we may assume that M^{•} is in fact a single locally finitely generated unit O_{F^{r},X}-module M. By proposition 6.7.5 there is a dense open subscheme U of Z (given its reduced induced structure) such that, if f denotes the immersion of U in X, then f^{!}M is a unit F^{r}-crystal. By proposition 5.10.1 there is a morphism

M → f_{+}f^{!}M

whose cone is supported on the complement of U in Z, which is of strictly lower dimension than Z, since U is dense.

Thus by the inductive hypothesis, to verify the proposition for M, it suffices to verify it for f_{+}f^{!}M. By proposition 4.2 we have

(6.8.1)  f_{+}f^{!}M ⊗^{𝕃}_{O_{X}} N^{•} = f_{+}(f^{!}M ⊗^{𝕃}_{O_{X}} f^{!}N^{•}[d_{X/U}]).

But f^{!}M is a unit F^{r}-crystal on U, so we may conclude by what we have already proved that

f^{!}M ⊗^{𝕃}_{O_{X}} f^{!}N^{•}[d_{X/U}]

is an object of D_{lfgu}^{b}(O_{F^{r},U}), whence so is its image after applying f_{+} by corollary 6.6.3. By (6.8.1) we are done. □

(6.9) Let X be a smooth k-scheme. If k′ is an extension field of k, and X′ is the base-change of X over k′, then base-change of O_{F^{r},X}-modules via k induces functors

k′ ⊗_{k} – : μ_{lfgu}(X) → μ_{lfgu}(X′)

and

k′ ⊗_{k} – : D_{lfgu}^{b}(O_{F^{r},X}) → D_{lfgu}^{b}(O_{F^{r},X′}).

Again, a case of particular interest is that in which k′ is a purely inseparable algebraic extension of k.

Proposition 6.9.1. Let k′ be a purely inseparable algebraic extension of k, let X be a smooth k-scheme, and let X′ = k′ ×_{k} X be the base-change of X over k′.

(i) For any two complexes M^{•} and N^{•} in D_{lfgu}^{b}(O_{F^{r},X}), the natural morphism

\underline{RHom}_{O_{F^{r} ,X}}^{•} (M^{•}, N^{•}) → \underline{RHom}_{O_{F^{r} ,X′}}^{•} (k′ ⊗_{k} M^{•}, k′ ⊗_{k} N^{•}) of (1.13.5) is an isomorphism.

(ii) The functor k′ ⊗_{k} – : μ_{lfgu}(X) → μ_{lfgu}(X′) induces an equivalence of categories.

As a consequence of (i) and (ii) we deduce that

k′ ⊗_{k} – : D_{lfgu}^{b}(O_{F^{r},X}) → D_{lfgu}^{b}(O_{F^{r},X′})

induces an equivalence of categories.

The statement of this proposition is similar to that of proposition 5.11.1, except that k is allowed to be an arbitrary inseparable extension of k.

Proof. To prove (i), we may use truncation and induction on the length of a complex to restrict to the case in which each of M^{•} and N^{•} are objects M and N of μ_{lfgu}(X). We may also work locally on X, and so we may assume that X is affine.

Then, as in the proof of proposition 6.5, we may find a finite length resolution P^{•} of M by O_{F^{r},X}-modules which are free of finite rank. We may use this resolution to rewrite the natural isomorphism of (i) as the map

\underline{RHom}_{O_{F^{r} ,X}}^{•} (M, N) = \underline{Hom}_{O_{F^{r} ,X}}^{•} (P^{•}, N)

→ \underline{Hom}_{O_{F^{r} ,X′}}^{•} (k′ ⊗_{k} P^{•}, k′ ⊗_{k} N) = \underline{RHom}_{O_{F^{r} ,X′}}^{•} (k′ ⊗_{k} M^{•}, k′ ⊗_{k} N).

(since k′ ⊗_{k} P^{•} is clearly a locally free resolution of k′ ⊗_{k} M; compare remark 1.13.2). Thus we must show that the morphism of complexes

\underline{Hom}_{O_{F^{r} ,X}}^{•} (P^{•}, N) → \underline{Hom}_{O_{F^{r} ,X′}}^{•} (k′ ⊗_{k} P^{•}, k′ ⊗_{k} N)

is a quasi-isomorphism.

Since each P^{•} is free of finite rank, and since N is generated by finitely many of its global sections (recall that we are assuming X is affine), by regarding k′ as the direct limit of its finite sub-extensions, we see that it enough to verify this when k′ is a finite extension of k. But the result now follows from proposition 5.11.1 (i).

The proof of part (ii) uses a similar argument. Since (i) shows that the functor in question is fully faithful, we need only show that any object M′ of μ_{lfgu}(X′) is isomorphic to k′ ⊗_{k} M for some object M of μ_{lfgu}(X). Since X is of finite type and M′ is determined by a generator with a coherent domain, by the same direct limit argument as used above, we may first reduce to the case in which k′ is finite over k, and then the result follows from 5.11.1 (ii). □

7. O_{F^{r}}-modules on the étale site

(7.1) We let X_{ét} denote the small étale site of X, and we let π_{X} : X_{ét} → X denote the natural morphism of sites. (In this context, we write X without a subscript to indicate the Zariski site of X.) As usual, O_{Xét} will denote the structure sheaf of X_{ét}, defined by O_{Xét}(U) = O_{U}(U). The morphism π_{X} is naturally a morphism of locally ringed sites.

We may use π_{X} to pull-back the sheaf O_{F^{r},X}, and for any U étale over X, there is a natural isomorphism π_{X}^{∗}O_{F^{r},X}(U) = O_{F^{r},U}(U). Thus π_{X}^{∗}O_{F^{r},X} is naturally a sheaf of rings on X_{ét}. (Note that this is not a formal consequence of it being the ringed site pull-back of a sheaf of rings, because O_{X} is not in the center of O_{F^{r},X}.)

Definition 7.1.1. We define O_{F^{r},Xét} = π_{X}^{∗}O_{F^{r},X}, with its natural structure as a sheaf of rings.

(7.2) The maps

O_{F^{r},U}(U) → Hom_{𝔽_{q}}(O_{U}(U), O_{U}(U))

induce a map

O_{F^{r},Xét} → Hom_{𝔽_{q}}(O_{Xét}, O_{Xét}).

The following lemma shows that O_{F^{r},Xét} always acts faithfully on O_{Xét}.

Lemma 7.2.1. The map

O_{F^{r},Xét} → Hom_{𝔽_{p}}(O_{Xét}, O_{Xét})

is injective.

Proof. Let k̄ denote the separable closure of k. It suffices to show that the map

A [F^{r}] → Hom_{𝔽_{q}}(A ⊗_{k} k̄, A ⊗_{k} k̄)

is injective, and for this it suffices to show that the map

A [F^{r}] → Hom_{𝔽_{q}}(k̄, A ⊗_{k} k̄)

is injective. But this map is obtained by tensoring the injective map k [F^{r}] → Hom_{𝔽_{q}}(k̄, k̄) on the left with A over k. □

(7.3) Just as we have considered O_{F^{r},X}-modules, we will now also have to consider O_{F^{r},Xét}-modules. In particular we may consider sheaves of left O_{F^{r},Xét}- modules on X_{ét} which are quasi-coherent as O_{Xét}-modules. We will refer to such a sheaf as a quasi-coherent left O_{F^{r},Xét}-module, and let μ(X_{ét}) denote the category of such sheaves.

If M is a quasi-coherent left O_{F^{r},X}-module, then π_{X}^{∗}M is naturally a quasicoherent left O_{F^{r},Xét}-module, and the natural map M → π_{X∗}π_{X}^{∗}M is an isomorphism. Conversely, the theory of étale descent shows that if N is a quasi-coherent left O_{F^{r},Xét}-module, then π_{X∗}N is a quasi-coherent left O_{F^{r},X}-module, and the natural map π_{X}^{∗}π_{X∗}N → N is an isomorphism. Thus the functors π_{X}^{∗} and π_{X∗} provide an equivalence of categories between μ(X) and μ(X_{ét}).

We let D^{b}(O_{F^{r},Xét}) denote the derived category of finite-length complexes of left O_{F^{r},Xét}-modules. We let D_{qc}^{b}(O_{F^{r},Xét}) denote the sub-triangulated category of D^{b}(O_{F^{r},Xét}) consisting of complexes whose cohomology sheaves are quasi-coherent left O_{F^{r},Xét}-modules. We let D^{b}(μ(X_{ét})) denote the derived category of finite length complexes of quasi-coherent left O_{F^{r},Xét}-modules. There are natural morphisms of triangulated categories

D^{b}(μ(X_{ét})) → D_{qc}^{b}(O_{F^{r},Xét}) → D^{b}(O_{F^{r},Xét}).

Since étale morphisms are flat, the ringed site pull-back π_{X}^{∗} from X to X_{ét} is exact, and so we get a commutative diagram of morphisms of triangulated categories

D^{b}(μ(X)) → D_{qc}^{b}(O_{F^{r},X}) → D^{b}(O_{F^{r},X})
↓ π_{X}^{∗} ↓ π_{X}^{∗} ↓ π_{X}^{∗}
D^{b}(μ(X_{ét})) → D_{qc}^{b}(O_{F^{r},Xét}) → D^{b}(O_{F^{r},Xét}).

Theorem 7.3.1. All four arrows in the square

D^{b}(μ(X)) → D_{qc}^{b}(O_{F^{r},X})
↓ π_{X}^{∗}  ↓ π_{X}^{∗}
D^{b}(μ(X_{ét})) → D_{qc}^{b}(O_{F^{r},Xét})

are equivalences of triangulated categories.

Proof. That the left vertical arrow is an equivalence follows from étale descent. That the two horizontal arrows are equivalences follows from Bernstein’s theorem [Bor, VI 2.10]. Thus the right vertical arrow is also an equivalence. (That Bernstein’s theorem holds in the étale case is a consequence of étale descent. More precisely, the crux of the proof of Bernstein’s theorem is that quasi-coherent cohomology vanishes on an affine scheme, and étale descent shows that quasi-coherent cohomology computed on the étale site agrees with quasi-coherent cohomology computed on the Zariski site.) □

We can describe explicitly a quasi-inverse of the equivalence of categories

π_{X}^{∗} : D_{qc}^{b}(O_{F^{r},X}) → D_{qc}^{b}(O_{F^{r},Xét}).

First note that by taking injective resolutions of objects in D_{qc}^{b}(O_{F^{r},X}) we may compute the derived functor

Rπ_{X∗} : D_{qc}^{b}(O_{F^{r},Xét}) → D_{qc}^{+}(O_{F^{r},X}).

Proposition 7.3.2. The functor Rπ_{X∗} restricts to a functor

Rπ_{X∗} : D_{qc}^{b}(O_{F^{r},Xét}) → D_{qc}^{b}(F^{r}, X_{ét}),

which is a quasi-inverse to the functor

π_{X}^{∗} : D_{qc}^{b}(O_{F^{r},X}) → D_{qc}^{b}(O_{F^{r},Xét}).

Proof. If M^{•} is a complex in D_{qc}^{b}(O_{F^{r},Xét}) there is an adjunction morphism

(7.3.3).  π_{X}^{∗}Rπ_{X∗}M^{•} → M^{•},

We will show that Rπ_{X∗}M^{•} is an object of D_{qc}^{b}(O_{F^{r},X}) and that the adjunction morphism is an isomorphism.

By Bernstein’s theorem, we may assume that M^{•} is a complex in D^{b}(μ(X_{ét})). The fact that the étale cohomology of a quasi-coherent sheaf on an affine scheme vanishes shows that the higher derived direct images of π_{X} vanish when applied to quasi-coherent left O_{F^{r},Xét}-modules, and thus when applied to complexes of quasicoherent left O_{F^{r},Xét}-modules. Thus Rπ_{X∗}M^{•} ⟶^{∼} π_{X∗}M^{•} is a bounded complex of quasi-coherent O_{F^{r},X}-modules and

π_{X}^{∗}Rπ_{X∗}M^{•} ⟶^{∼} π_{X}^{∗}π_{X∗}M^{•} ⟶^{∼} M^{•},

the second natural isomorphism holding by étale descent. The composite of these morphisms is the adjunction morphism (7.3.3), and so we see that this morphism is indeed a natural isomorphism.

A similar argument shows that for any complex M^{•} in D_{qc}^{b}(O_{F^{r},X}) the adjunction morphism M^{•} → Rπ_{X∗}π_{X}^{∗}M^{•} is an isomorphism.

Thus we see that indeed Rπ_{X∗} takes D_{qc}^{b}(O_{F^{r},Xét}) to D_{qc}^{b}(O_{F^{r},X}), and is a quasi-inverse to π_{X}^{∗}. □

(7.4) We now briefly discuss pull-back and push-forwards of O_{F^{r}}-modules in the context of the étale site.

Definition 7.4.1. If f : Y → X is a morphism of smooth k-schemes then we define the (O_{F^{r},Yét}, f^{−1}O_{F^{r},Xét})-bimodule O_{F^{r},Yét→Xét} as

O_{F^{r},Yét→Xét} = π_{Y}^{∗} O_{F^{r},Y →X},

where the bimodule structure is induced by the (O_{F^{r},Y} , f^{−1}O_{F^{r},X})-bimodule structure on O_{F^{r},Y →X}.

Definition 7.4.2. Let f : Y → X be a morphism of smooth connected k-schemes. We define f^{!} : D_{qc}^{b}(O_{F^{r},Xét}) → D_{qc}^{b}(O_{F^{r},Yét}) as follows:

f^{!}M^{•} = O_{F^{r},Yét→Xét} ⊗^{𝕃}_{f^{−1}O_{Xét}} f^{−1}M[d_{Y/X}].

Just as in the case of Zariski O_{F^{r}}-modules, we must show that this functor is welldefined. But this follows from the following theorem (and so ultimately from étate descent), comparing pull-back for Zariski and étale O_{F^{r}}-modules, and the fact that the pull-back for Zariski O_{F^{r}}-modules is well defined.

Proposition 7.4.3. Let f : Y → X be a morphism of smooth k-schemes, and let π_{X} : X_{ét} → X and π_{Y} : Y_{ét} → Y denote the natural morphisms of sites. Then there are natural isomorphisms Rπ_{Y ∗}f^{!} ⟶^{∼} f^{!}Rπ_{X∗} and π_{Y}^{∗} f^{!} ⟶^{∼} f^{!}π_{X}^{∗}.

Proof. Since Rπ_{X∗} and π_{X}^{∗} (respectively Rπ_{Y ∗} and π_{Y}^{∗}) are quasi-inverse, it is enough to construct the second isomorphism.

If M^{•} is a complex in D_{qc}^{b}(O_{F^{r},X}), then we have (using definition 7.4.1)

π_{Y}^{∗} f^{!}M^{•} = π_{Y}^{∗} (O_{F^{r},Y} ⊗_{f^{−1}O_{F^{r} ,X}} f^{−1}M^{•})

⟶^{∼} O_{F^{r},Yét} ⊗_{f^{−1}O_{F^{r} ,Xét}} f^{−1}π_{X}^{∗}M^{•} = f^{!}π_{X}^{∗}M^{•}.

□

Definition 7.4.4. If f : Y → X is a morphism of smooth k-schemes then we define the (f^{−1}O_{F^{r},Xét}, O_{F^{r},Yét})-bimodule O_{F^{r},Xét←Yét} as

O_{F^{r},Xét←Yét} = π_{Y}^{∗} O_{F^{r},X←Y} ,

with the bimodule structure induced by the (f^{−1}O_{F^{r},X}, O_{F^{r},Y})-bimodule structure on O_{F^{r},X←Y} .

Definition 7.4.5. Let f : Y → X be a morphism of smooth k-schemes. We define

f_{+} : D_{qc}^{b}(μ(Y_{ét})) → D_{qc}^{b}(μ(X_{ét}))

by the formula

f_{+}M^{•} = Rf_{∗}(O_{F^{r},Xét←Yét} ⊗^{𝕃}_{O_{F^{r} ,Yét}} M^{•}).

As in the case of Zariski O_{F^{r}}-modules, we must show that this functor is welldefined. Just as for pull-backs, this will follow from étale descent, and the corresponding result for Zariski O_{F^{r}}-modules:

Proposition 7.4.6. Let f : Y → X be a morphism of smooth k-schemes, and let π_{X} : X_{ét} → X and π_{Y} : Y_{ét} → Y denote the natural morphisms of sites. Then there are natural isomorphisms Rπ_{X∗}f_{+} = f_{+}Rπ_{Y ∗} and π_{X}^{∗}f_{+} = f_{+}π_{Y}^{∗} .

Proof. Since Rπ_{X∗} and π_{X}^{∗} (respectively Rπ_{Y ∗} and π_{Y}^{∗}) are quasi-inverse, it is enough to construct the first isomorphism.

If M^{•} is a complex in D_{qc}^{b}(O_{F^{r},Yét}), then we have (using definition 7.4.4)

f_{+}Rπ_{Y ∗}M^{•} = Rf_{∗}(O_{F^{r},X←Y} ⊗^{𝕃}_{O_{F^{r} ,Y}} Rπ_{Y} ∗M^{•})

⟶^{∼} Rf_{∗}Rπ_{Y ∗}(O_{Fét,Xét←Yét} ⊗^{𝕃}_{O_{F^{r} ,Yét}} M^{•}) ⟶^{∼} Rπ_{X∗}f_{+}M^{•}.

□

Propositions 7.3.2, 7.4.3 and 7.4.6 show that the morphisms π_{X}^{∗} and Rπ_{X∗} provide an equivalence of categories which commutes with pull-backs and pushforwards. We now observe that all the results proved for O_{F^{r},X}-modules have obvious analogues for O_{F^{r},Xét}-modules. §4, §5 and §6 have analogues for O_{F^{r},Xét}- modules. For most of the results, this follows immediately by étale descent. The only exceptions are the results treating \underline{RHom}_{O_{F^{r} ,X}}^{•} (–, –), since this functor produces sheaves of 𝔽_{q}-modules which are in general not quasi-coherent and so do not satisfy étale descent. However, these results are also easily established simply by working in the étale context from the beginning.

8. 𝔽_{q}-sheaves on the étale site

(8.1) Recall [SGA 4] that a constructible sheaf of 𝔽_{q}-modules on the étale site of a scheme X is a sheaf of 𝔽_{q}-sheaves on X_{ét} which restricts to a locally constant sheaf of finite rank 𝔽_{q}-vector spaces along the members of some stratification of X by locally closed subsets.

We let D^{+}(X_{ét}) denote the derived category of complexes of 𝔽_{q} sheaves on X_{ét} which are bounded below. We let D_{c}^{b}(X_{ét}) denote the full triangulated subcategory of D^{+}(X_{ét}) consisting of finite-length complexes of 𝔽_{q}-sheaves on X_{ét} whose cohomology sheaves are constructible.

(8.2) If f : Y → X is a morphism of schemes, it induces morphisms

f^{−1} : D^{+}(X_{ét}) → D^{+}(Y_{ét}) and Rf_{∗} : D^{+}(Y_{ét}) → D^{+}(X_{ét}).

If f is an immersion, we also have the extension by zero functor

f_{!} : D^{+}(Y_{ét}) → D^{+}(X_{ét}).

If f : Y → X is an arbitrary morphism of k-schemes, let i : Y → Ȳ be an open immersion of Y into a proper k scheme. Such an immersion exists by Nagata’s theorem. We may factor f into the product of an immersion and a proper morphism, as follows:

Y ⟶^{Γ_{f}} X × Y ⟶^{id_{X}×i} X × Ȳ ⟶^{p_{1}} X,

where Γ_{f} is the graph of f, which is an immersion, and p_{1} is the projection onto the first factor, which is proper.

We define

f_{!} = Rp_{1∗} ((id_{X} × i)Γ_{f})_{!} : D^{+}(Y_{ét}) → D^{+}(X_{ét}).

If f = hj is any other factorization of f into the composition of a proper map h with an immersion j then one verifies that f_{!}, as defined above, is naturally isomorphic to the functor Rh_{∗}j_{!}. Thus the functor f_{!} is (up to natural isomorphism) independent of the choice of factorization of f as the composition of an immersion and a proper map. (See [De, p. 48]).

If g : Z → Y is a second morphism of k-schemes, then there are natural isomorphisms

(fg)^{−1} = g^{−1}f^{−1} and (fg)_{!} = f_{!}g_{!}.

(8.3) Now any morphism f : Y → X pulls back a stratification on X to a stratification on Y, and f^{−1} is an exact functor which takes locally constant sheaves to locally constant sheaves. Thus f^{−1} : D_{c}^{b}(X_{ét}) → D_{c}^{b}(Y_{ét}).

If f : Y → X is an immersion, then f_{!} is an exact functor, which takes constructible sheaves on Y to constructible sheaves on X whose stalks are zero outside Y. Thus f_{!} : D_{c}^{b}(Y_{ét}) → D_{c}^{b}(X_{ét}). If f : Y → X is a proper morphism, then it is a consequence of the proper base-change theorem that Rf)∗ restricts to a functor (denoted by the same symbol) Rf_{∗} : D_{c}^{b}(Y_{ét}) → D_{c}^{b}(X_{ét}). Thus for any morphism f : Y → X of k-schemes, f_{!} restricts to a functor (denoted by the same symbol) f_{!} : D_{c}^{b}(Y_{ét}) → D_{c}^{b}(X_{ét}).

(8.4) We will frequently use the following analogue of proposition 6.7.5: if k is perfect and L is a constructible 𝔽_{q}-sheaf on X_{ét} for some smooth k-scheme X, with support contained in a closed subset Z of X, then there exists an open subscheme U of Z (given its reduced induced structure) which is smooth as a k-scheme, and such that the restriction of L to U is locally constant.

(8.5) In order to apply (8.4), we will have to be able replace k by its perfect closure if necessary. That this is legitimate follows from the topological invariance of the étale site. More precisely and more generally, let k′/k be a purely inseparable algebraic field extension. If X is any smooth k-scheme then the base-change X′ = k′ ×_{k} X is a smooth k′ scheme, and the morphism X′ → X induces an isomorphism between X_{ét} and X′_{ét}. Thus there is induced an equivalence of categories between D^{•}(X_{ét}) and D^{•}(X′_{ét}) (where • can be any one of +, −, b or ∅), which restricts to an equivalence of categories between D_{c}^{b}(X_{ét}) and D_{c}^{b}(X′_{ét}).

9. the functor Sol_{ét}

(9.1) If X is a smooth k-scheme, recall that D^{+}(X_{ét}) denotes the derived category of bounded below complexes of 𝔽_{q}-sheaves on X_{ét}.

Definition 9.2. Let X be a smooth k-scheme. Define

Sol_{ét}(–) = \underline{RHom}_{O_{F^{r} ,Xét}}^{•} (–, O_{Xét})[d_{X}] : D_{lfgu}^{b}(O_{F^{r},Xét}) → D^{b}(X_{ét}).

That this functor has image in D^{b}(X_{ét}) follows from proposition 5.12.

Remark 9.2.1 Note that the functor Sol_{ét} is compatible with inseparable basechange, in the sense that if k′ is a purely inseparable algebraic extension of k, if X is a smooth k-scheme, and if X′ is the base-change of X over k′, then the diagram

\xymatrix{D_{lfgu}^{b}(O_{F^{r},X}) \ar[r]^{{k′⊗_{k}– \atop ∼}} \ar[d]^{Solét} & D_{lfgu}^{b}(O_{F^{r},X′}) \ar[d]^{Solét} \\ D^{b}(X_{ét}) \ar[r]^{∼} & D^{b}(X′_{ét})}

commutes (by proposition 6.9.1).

Proposition 9.3. Let f : Y → X be a morphism of smooth k-schemes. Then the functors f^{−1} Sol_{ét} and Sol_{ét} f^{!} are naturally isomorphic, in a manner compatible with inseparable base-change.

Proof. If we note that f^{!}O_{Xét} = O_{Yét}[d_{Y/X}], then we may use proposition 2.6 to define a natural transformation

f^{−1} Sol_{ét}(M^{•}) = f^{−1}\underline{RHom}_{O_{F^{r} ,Xét}}^{•} (M^{•}, O_{Xét})[d_{X}]
→ \underline{RHom}_{O_{F^{r} ,Yét}}^{•} (f^{!}M^{•}, O_{Yét}[d_{Y/X}])[d_{X}]
= \underline{RHom}_{O_{F^{r} ,Yét}}^{•} (f^{!}M, O_{Yét})[d_{Y}]
= Sol_{ét}(f^{!}M^{•}).

We must now prove that this natural transformation is an isomorphism.

By lemma 2.7 this isomorphism is compatible with inseparable base-change. Thus we may replace k by its perfect closure, and for the duration of the proof we assume that k is perfect. Secondly, we may work locally on X and Y, and so we may assume that X is affine. Thirdly, truncation and induction on the length of the complex M^{•} allow us to reduce to the case of a single locally finitely generated unit O_{F^{r},Xét}-module M. This completes our initial set of reductions.

Let β : M → F_{X}^{r}^{∗}M be a generator of M, with M a coherent O_{Xét}-module. As in the proof of part (i) of theorem 5.6, we choose a resolution P^{•} of M by finite rank free O_{Xét}-modules, and a lift β^{•} : P^{•} → F_{X}^{r}^{∗}P^{•} of the morphism β. Let P^{i} be the locally finitely generated unit O_{F^{r},X}-module generated by β^{i}. Then the complex P^{•} is a resolution of M, and the complex f^{!}P^{•} is a resolution of f^{!}M.

Thus there is a spectral sequence computing the cohomology sheaves of Sol_{ét}(M) whose E_{1} terms are E_{1}^{i,j} = H^{j}(Sol_{ét}(P^{i})); applying the exact functor f^{−1} yields a spectral sequence computing the cohomology sheaves of f^{−1} Sol_{ét}(M) whose E_{1} terms are E_{1}^{i,j} = H^{j}(f^{−1} Sol_{ét}(P^{i})). Similarly, there is a spectral sequence computing the cohomology sheaves of H^{i}(Sol_{ét}(f^{!}M)) whose E_{1} terms are E_{1}^{i,j} = H^{j}(Sol_{ét}(f^{!}P^{i})). The natural transformation f^{−1} Sol_{ét} ↦ Sol_{ét} f^{!} induces a morphism of spectral sequences. Thus it suffices to show that the morphism

f^{−1} Sol_{ét}(P^{i}) → Sol_{ét} f^{!}P^{i}

is an isomorphism for each i. Write P = P^{i}, and chose an isomorphism P ⟶^{∼} O_{Xét}^{n}. Then the map β_{i} is identified with an n × n matrix μ

μ = β^{i} : O_{Xét}^{n} → O_{Xét}^{n}.

The generator P of P induces a free presentation of P

0 → O_{F^{r},Xét}^{n} →^{1−μF^{r}} O_{F^{r},Xét}^{n} → P → 0.

(In this diagram, the n×n matrix μF^{r} is acting by right multiplication on O_{F^{r},Xét}^{n}.) Similarly, f^{!}P is the unit O_{F^{r},Yét}-module with free presentation

0 → O_{F^{r},Yét}^{n} →^{1−μF^{r}} O_{F^{r},Yét}^{n} → f^{!}P → 0,

placed in degree d_{X/Y}.

We have the isomorphisms

Sol_{ét}(P) ⟶^{∼} \underline{Hom}_{O_{F^{r} ,Xét}}^{•} (O_{F^{r},Xét}^{n} →^{1−μF^{r}} O_{F^{r},Xét}^{n}, O_{Xét})[d_{X}]

⟶^{∼} (O_{Xét}^{n} →^{1−μF^{r}} O_{Xét}^{n})

(where in this last complex, the two members of the complex sit in degrees −d_{X} and −d_{X} + 1). If (y_{i}) is an n-tuple of sections of O_{X} over any étale neighborhood U of X, the equation

(1 − μF^{r})(x_{i}) = (x_{i}) − μ(x_{i}^{q}) = (y_{i})

describes a (not necessarily finite) étale cover of U, and so the morphism

1 − μF^{r} : O_{Xét}^{n} → O_{Xét}^{n}

is surjective. Thus Sol_{ét}(P) is isomorphic to the kernel of the morphism 1 − μF, placed in degree −d_{X}, and since f^{−1} is exact, we find that

f^{−1} Sol_{ét}(P) ⟶^{∼} f^{−1}ker(O_{Xét}^{n} →^{1−μF^{r}} O_{Xét}^{n})[d_{X}].

A similar computation shows that Sol_{ét}(f^{!}P) ⟶^{∼} ker(O_{Yét}^{n} →^{1−μF^{r}} O_{Yét}^{n})[d_{X}], and that the map f^{−1} Sol_{ét}(P) → Sol_{ét}(f^{!}P) is simply the natural map

f^{−1}ker(O_{Xét}^{n} →^{1−μF^{r}} O_{Xét}^{n})[d_{X}] → ker(O_{Yét}^{n} →^{1−μF^{r}} O_{Yét}^{n})[d_{X}].

Thus to prove the proposition, we are reduced to showing that this map is an isomorphism. We begin by studying a number of cases.

(i) f is the closed immersion of a divisor: As remarked at the beginning of the proof, we may work locally. Thus we may assume that Y is the divisor cut out by a global section a of O_{Xét}.

We may work on the level of stalks; thus we are reduced to showing that if A is a strictly Henselian local ring occurring as a stalk of a geometric point of X, μ is an n × n matrix with entries in A, and a ∈ A lies in the maximal ideal of A, then the natural map

ker(A^{n} →^{1−μF^{r}} A^{n}) → ker((A/a)^{n} →^{1−μF^{r}} (A/a)^{n})

is an isomorphism.

To see that it is injective, suppose that (ax_{i}) ∈ aA^{n} satisfies (1 − μF^{r})(ax_{i}) = 0. Then (ax_{i}) = μF^{r}(ax_{i}) = a^{q}μF^{r}(x_{i}), and

(x_{i}) = a^{q−1}μF^{r}(x_{i}) = a^{q−1}μF^{r}(a^{q−1}μF^{r}(x_{i}))

= a^{(q+1)(q−1)}μF^{r}μF^{r}(a^{q−1}μF^{r}(x_{i})) = …

Continuing, we find that the x_{i} ∈ A are divisible by arbitrarily high powers of a, and so vanish, since a lies in the maximal ideal of A. This yields the injectivity.

To see the surjectivity, suppose that (x̄_{i}) ∈ (A/a)^{n} satisfies (1 − μF^{r})(x̄_{i}) = 0. Let (x_{i}) ∈ A^{n} lift (x̄_{i}). Then (1−μF^{r})(x_{i}) = (ax′_{i}) for some (x′_{i}) ∈ A^{n}. Now since A is strictly Henselian, we may find an n-tuple (y_{i}) ∈ A^{n} solving (1 − a^{q−1}μF^{r}(y_{i})) = (x′_{i}). Then x_{i} − ay_{i} = x̄_{i} modulo the maximal ideal of A, and

(1 − μF^{r})(x_{i} − ay_{i}) = (ax′_{i} − ax′_{i}) = 0.

(ii) f is étale: We must show that

f^{−1}ker(O_{Xét}^{n} →^{1−μF^{r}} O_{Xét}^{n})[d_{X}] → ker(O_{Yét}^{n} →^{1−μF^{r}} O_{Yét}^{n})[d_{X}]

is an isomorphism. But since f is étale, f^{−1}O_{Xét} = O_{Yét}, and so

f^{−1}ker(O_{Xét}^{n} →^{1−μF^{r}} O_{Xét}^{n})[d_{X}] = ker(f^{−1}O_{Xét}^{n} →^{1−μF^{r}} f^{−1}O_{Xét}^{n})[d_{X}]

= ker(O_{Yét}^{n} →^{1−μF^{r}} O_{Yét}^{n})[d_{X}].

(Here we have used the exactness of f^{−1}.) Thus in this case we do indeed have an isomorphism.

(iii) f is the projection X × 𝔸_{k}^{n} → X: Let s be any section of f. Then the composite

s^{−1}f^{−1}ker(O_{Xét}^{n} →^{1−μF^{r}} O_{Xét}^{n})[d_{X}] → s^{−1}ker(O_{Yét}^{n} →^{1−μF^{r}} O_{Yét}^{n})[d_{X}]

→ ker(O_{Xét}^{n} →^{1−μF^{r}} O_{Xét}^{n})[d_{X}],

corresponds to the identity map, and the second arrow factors as a composite of embeddings of smooth divisors, and so is an isomorphism by (i). Thus the first arrow is also an isomorphism. It follows that

f^{−1}ker(O_{Xét}^{n} →^{1−μF^{r}} O_{Xét}^{n})[d_{X}] → ker(O_{Yét}^{n} →^{1−μF^{r}} O_{Yét}^{n})[d_{X}]

is an isomorphism, as it is after pulling back by any section.

More precisely, denote the cone of this morphism by C. Any closed point of X × 𝔸_{k}^{n} is in the image of an étale local section of f (since k is perfect), and so we see that C has vanishing stalks at any closed point of X × 𝔸_{k}^{n}. Since these elements are dense in X × 𝔸_{k}^{n}, we see that C does indeed vanish.

(iv) f is arbitrary: We factor f : Y → X as

Y →^{Γ_{f}} X × Y →^{p_{1}} X.

Now Γ_{f} is a closed immersion, and so locally factors as a composite of embeddings of smooth divisors. Thus part (i) above shows that the proposition holds for Γ_{f}.

Since the morphism Y → Spec k is smooth, it factors locally as an étale morphism followed by the map 𝔸_{k}^{n} → Spec k, and so the result follows by (ii) and (iii). □

Example 9.3.1 If X is a smooth k-scheme, then Sol_{ét}(O_{Xét}) = 𝔽_{q} [d_{X}]. This result follows from proposition 9.3, since taking f : X → Spec k to be the projection to the point, one sees that it suffices to prove the result for the point, where it is immediately verified.

One may also verify this result directly by computing

Sol_{ét}(O_{Xét}) = \underline{RHom}_{O_{F^{r} ,Xét}}^{•} (O_{Xét}, O_{Xét})[d_{X}]

using the canonical resolution of O_{Xét} by left O_{F^{r},Xét}-modules:

0 ⟶ O_{F^{r},Xét} ⟶^{1−F^{r}} O_{F^{r},Xét}.

It then amounts to the fact that the sequence of sheaves

0 ⟶ 𝔽_{q} ⟶ O_{Xét} ⟶^{1−F^{r}} O_{Xét} → 0

is exact on X_{ét}.

(9.4) The following lemma is a particular instance of [Ka, Prop. 4.1.1]. We recall the proof, which depends on a technique of Lang.

Lemma 9.4.1. If X is a smooth k-scheme and M is a unit F^{r}-crystal on X, then M is, étale locally on X, isomorphic as a left O_{F^{r},X}-module to O_{X}^{n}, for some n.

Proof. By assumption (and lemma 6.7.1) M is locally free of finite rank on X. Let U be any open set of X over which M ⟶^{∼} O_{X}^{n} (as O_{X}-modules). Then the structural morphism of M is an isomorphism O_{X}^{n} = F_{X}^{r}^{∗}O_{X}^{n} → O_{X}^{n}, and so is given by some section A of GL_{n}(O_{X}) over U.

If B is a section of GL_{n}(O_{X}), let B^{[q]} denote the section of GL_{n}(O_{X}) obtained by raising the entries of B to the q^{th} power. Lang observes that the equation B^{[q]} = BA describes a finite étale cover V of U. Thus over V, we obtain the following commutative diagram:

\xymatrix{O_{V}^{n} \ar[r]^{A} \ar[dr]_{B^{[q]}} & O_{V}^{n} \ar[d]^{B} \\  & O_{V}^{n} ,}

which shows that M is isomorphic as a left O_{F^{r},X}-module to O_{X}^{n} over V. □

Corollary 9.4.2. If X is a smooth k-scheme and M is a unit F^{r}-crystal on X_{ét} then \underline{RHom}_{O_{F^{r} ,Xét}}^{•} (M, O_{Xét}) = L for some locally constant étale sheaf L. Furthermore, the natural transformation

M → \underline{Hom}_{𝔽_{q}}(L, O_{Xét}) ⟶^{∼} \underline{Hom}_{𝔽_{q}}(L, 𝔽_{q}) ⊗_{𝔽_{q}} O_{Xét}

is an isomorphism.

Proof. Both claims may be checked locally on X_{ét}, so that lemma 9.4.1 reduces us to checking them for M = O_{X}^{n}, in which case they follow by Example 9.3.1. □

Proposition 9.5. (i) If f : Y → X is an open immersion of smooth k-schemes, there is a natural isomorphism of functors

f_{!} Sol_{ét} ⟶^{∼} Sol_{ét} f_{+}.

(ii) If g : Z → Y is a second open immersion of smooth k-schemes then the diagram of natural isomorphisms

\xymatrix{Sol_{ét} f_{+}g_{+} \ar[r]^{∼}_{(3.8)} \ar[d]^{∼}_{part (i)} & Sol_{ét}(fg)_{+} \ar[dd]_{part (i)}^{∼} \\ f_{!} Sol_{ét} g_{+} \ar[d]^{∼}_{part (i)} \\ f_{!}g_{!} Sol_{ét} \ar[r]^{∼} & (fg)_{!} Sol_{ét}}

(in which the natural isomorphisms have been labelled by the results which give rise to them) commutes.

Proof. We first construct a natural transformation f_{!} Sol_{ét} → Sol_{ét} f_{+}, which will clearly satisfy the compatibility with compositions claimed by part (ii), and then show it is an isomorphism.

Recall from lemma 4.3.1 that f^{!}O_{Xét} = f^{−1}O_{Xét} = O_{Uét}, and that if M^{•} is any complex in D_{lfgu}^{b}(O_{F^{r},Uét}), there is a natural isomorphism f^{!}f_{+}M^{•} ⟶^{∼} M^{•}. Thus we obtain the natural isomorphism

f^{−1} Sol_{ét}(f_{+}M^{•}) = f^{−1}\underline{RHom}_{O_{F^{r} ,Xét}}^{•} (f_{+}M^{•}, O_{Xét})[d_{X}]

⟶^{{(1) \atop ∼}} \underline{RHom}_{O_{F^{r} ,Uét}}^{•} (f^{!}f_{+}M^{•}, O_{Uét})[d_{U}]

⟶^{∼} \underline{RHom}_{O_{F^{r} ,Uét}}^{•} (M^{•}, O_{Uét})[d_{U}] = Sol_{ét}(M^{•}),

in which the isomorphism labelled (1) is provided by proposition 2.6. By adjointness of f_{!} and f^{−1}, the inverse of this isomorphism induces a morphism

f_{!} Sol_{ét}(M^{•}) → Sol_{ét}(f_{+}M^{•}),

which is the required natural transformation. It is straightforward to verify that this natural transformation satisfies the claim of part (ii).

Let Z = X \ Y denote the complement of Y, endowed with the reduced induced scheme structure, and let g : Z → X be the closed immersion of Z into X. To show that the morphism f_{!} Sol_{ét}(M^{•}) → Sol_{ét}(f_{+}M^{•}) is an isomorphism, it suffices to show that g^{−1} Sol_{ét}(f_{+}M^{•}) is quasi-isomorphic to zero.

This can be checked locally on X = Spec A, and so we may assume that X is affine, and that Z = V (a_{1}, …, a_{s}). Also, by truncation and induction on the length of a complex, we may reduce to the case of a single finitely generated unit O_{F^{r},Y} -module M. In this situation, the proof of corollary 6.6.3 shows that f_{+}M is represented by a Čech complex whose members are finite direct sums of modules of the form j_{+}N, where j is the open immersion of some open subscheme of Y of the form X_{a} (with a being a product of some number of the a_{i}’s) into X, and N is the restriction of M to X_{a}. The cohomology sheaves of Sol_{ét}(f_{+}M) are computed by a spectral sequence whose E_{1} terms are finite direct sums of the cohomology sheaves of the Sol_{ét}(j_{+}N). Thus g^{−1} Sol_{ét}(f_{+}M) is computed by pulling back this spectral sequence by the exact functor g^{−1}, and so it suffices to show that each g^{−1} Sol_{ét}(j_{+}N) is quasi-isomorphic to zero. Let h : V (a) → X be the closed immersion. Then Z ⊂ V (a), and so it will in particular suffice to show that h^{−1} Sol_{ét}(j_{+}N) is quasi-isomorphic to zero. Thus we need only consider the case where Y is the complement of a divisor V (a) in X.

The proof of proposition 6.7.1 shows that f_{+}M is a locally finitely generated unit O_{F^{r},X}-module which admits a generator of the form M →^{β} F_{X}^{r}^{∗}M, where M is a coherent O_{Xét}-module, and β = aβ′ for some β′ : M → F_{X}^{r}^{∗}M.

Let P be a resolution of M by free finite rank O_{Xét}-modules, and lift β′ to

β′^{•} : P → F_{X}^{r}^{∗}P.

Define β^{•} = aβ′^{•}. Let P^{i} denote the unit O_{F^{r},Xét}-module generated by β^{i}. Then P^{•} is a resolution of M, and by the usual spectral sequence argument, it suffices to show that g^{−1} Sol_{ét}(P^{i}) is quasi-isomorphic to zero for each i.

Let P^{i} = O_{Xét}^{n} and write

μ = β′^{i} : O_{Xét}^{n} → F_{X}^{r}^{∗}O_{Xét}^{n} = O_{Xét}^{n}.

Then, as was observed in the proof of proposition 9.3,

Sol_{ét}(P^{i}) = ker(O_{Xét}^{n} →^{1−aμF^{r}} O_{Xét}^{n})[d_{X}].

We want to show that the stalks of this kernel vanish at the geometric points of V (a). Let A be the stalk of O_{Xét} at such a point; then a lies in the maximal ideal of A. Thus if (x_{i}) ∈ A^{n} lies in this kernel,

(x_{i}) = aμF^{r}(x_{i}) = a^{q+1}μ(F^{r}μF^{r}(x_{i})) = ⋯;

thus each x_{i} is divisible by arbitrarily high powers of a and hence it vanishes. This completes the proof of the proposition. □

Remark 9.5.1 The reader should compare this result with lemma 3.3 of [De]. In that lemma Deligne takes a “dual” approach to the one taken here. Suppose that f : Y → X is an open immersion of the complement of a divisor, and that M is a unit F^{r}-crystal on Y. Let us follow the steps of the proof of proposition 6.7.1 by letting N be a coherent extension of M to X. In this proof we tensor the coherent sheaf N with large positive powers of the sheaf O_{X}(D) to obtain a generator

γ : N ⊗_{O_{X}} O_{X}(D)^{⊗m} → F_{X}^{r∗}(N ⊗_{O_{X}} O_{X}(D)^{⊗m})

for f_{+}M. Rather than doing this, Deligne tensors N with a large negative power of O_{X}(D), to obtain a morphism

Φ : F_{X}^{r∗}(N ⊗_{O_{X}} O_{X}(−D)^{⊗m}) → N ⊗_{O_{X}} O_{X}(−D)^{⊗m}.

Thus N becomes a (non-unit) O_{F^{r},Xét}-module, and Deligne shows that

\underline{RHom}_{O_{F^{r} ,Xét}}^{•} (O_{X}, N ⊗_{O_{X}} O_{X}(−D)^{⊗m}) = j_{!}\underline{RHom}_{O_{F^{r} ,Yét}}^{•} (O_{Y} , M).

The disadvantage of this approach is that there is no unique choice of the integer m, and so the O_{F^{r},X}-module N ⊗_{O_{X}} O_{X}(−D)^{⊗m} is not uniquely determined. From our point of view, this is analogous to the fact that there is no unique generator for a unit O_{F^{r},X}-module. Although the generators are useful for computations, it is the locally finitely generated unit O_{F^{r},X}-module f_{+}M which is the canonical object associated with pushing forward M via f.

Proposition 9.6. (i) Let f : Y → X be a proper morphism of smooth k-schemes. Then there is a natural isomorphism

Sol_{ét} f_{+} ⟶^{∼} f_{!} Sol_{ét} .

(ii) If g : Z → Y is a second proper morphism of smooth k-schemes then the diagram of natural isomorphisms

\xymatrix{Sol_{ét} f_{+}g_{+} \ar[r]^{∼}_{(3.8)} \ar[d]^{∼}_{part (i)} & Sol_{ét}(fg)_{+} \ar[dd]_{part (i)}^{∼} \\ f_{!} Sol_{ét} g_{+} \ar[d]^{∼}_{part (i)} \\ f_{!}g_{!} Sol_{ét} \ar[r]^{∼} & (fg)_{!} Sol_{ét}}

(in which the natural isomorphisms have been labelled by the results which give rise to them) commutes.

Proof. Since f is a proper morphism, we have f_{!} = Rf_{∗}. Let M^{•} be an object of D_{lfgu}^{b}(O_{F^{r},Y}). Then using part (i) of theorem 4.4.1, we compute that

Sol_{ét}(f_{+}M^{•}) = \underline{RHom}_{O_{F^{r} ,Xét}}^{•} (f_{+}M^{•}, O_{F^{r},Xét})[d_{X}]

⟶^{∼} Rf_{∗}\underline{RHom}_{O_{F^{r} ,Yét}}^{•} (M^{•}, f^{!}O_{F^{r},Xét})[d_{X}] ⟶^{∼} f_{!}\underline{RHom}_{O_{F^{r} ,Yét}}^{•} (M^{•}, O_{F^{r},Yét})[d_{Y}]

= f_{!} Sol_{ét} M^{•}.

The compatibility with compositions claimed by part (ii) follows from the given construction, together with part (iii) of theorem 4.4.1. □

(9.7) We would like to prove an analogue of propositions 9.5 and 9.6 for an arbitrary morphism f : Y → X of smooth k-schemes. If we can factor f as a composition of open immersions and proper morphisms than we can combine the results of propositions 9.5 and 9.6 to obtain an analogous result for the morphism f. Unfortunately, in order to prove that this isomorphism is independent of the choice of such a factorization, our techniques require us to assume that f has a factorization of a more restricted type.

We will say that f is allowable if there is a smooth k-scheme W and a factorization f = gh, where g : Y → W is an immersion and h : W → X is a smooth proper morphism. The main examples of allowable morphisms that we have in mind are quasi-projective morphisms. If one had resolution of singularities in characteristic p then by combining it with Nagata’s theorem on the existence of compactifications, one could show that every morphism f is allowable.

Theorem 9.7.1. (i) Let f : Y → X be an allowable morphism of smooth kschemes. Then there is a natural isomorphism

Sol_{ét} f_{+} ⟶^{∼} f_{!} Sol_{ét} .

(ii) If g : Z → Y is a second allowable morphism of smooth k-schemes such that the composite fg is also allowable then the diagram of natural isomorphisms

\xymatrix{Sol_{ét} f_{+}g_{+} \ar[r]^{∼}_{(3.8)} \ar[d]^{∼}_{part (i)} & Sol_{ét}(fg)_{+} \ar[dd]_{part (i)}^{∼} \\ f_{!} Sol_{ét} g_{+} \ar[d]^{∼}_{part (i)} \\ f_{!}g_{!} Sol_{ét} \ar[r]^{∼} & (fg)_{!} Sol_{ét}}

(in which the natural isomorphisms have been labelled by the results which give rise to them) commutes.

(iii) The natural transformation of (i) is compatible with inseparable base-change.

(9.7.2) The construction of the isomorphism of part (i) will depend upon the factorization of the allowable morphism f into a composition of proper morphisms and open immersions, and so not only is the compatibility with respect to compositions as stated in part (ii) not immediate from the construction, but it implicitly includes as a special case the independence of the construction of part (i) from the particular choice of such a factorization.

Lemma 9.7.3. Suppose that f : Y → X is a morphism of smooth k-schemes which has two factorizations,

\xymatrix{f : Y \ar[r]^{j_{1}} & W_{1} \ar[r]^{p_{1}} & X,}

and

\xymatrix{f : Y \ar[r]^{p_{2}} & W_{1} \ar[r]^{j_{2}} & X,}

in which each of W_{1}, W_{2} is a smooth k-scheme, each of the morphisms j_{1}, j_{2} is an open immersion, and each of the morphisms p_{1}, p_{2} is proper. Then the diagram of natural isomorphisms

\xymatrix{p_{1+}j_{1+} Sol_{ét} \ar[d]_{(3.8)}^{∼} \ar[r]^{∼}_{(9.5)} & p_{1+} Sol_{ét} j_{1!} \ar[r]^{∼}_{(9.6)} & Sol_{ét} p_{1!}j_{1!} \ar[d]^{∼} \\ f_{+} Sol_{ét} &  & Sol_{ét} f_{!} \\ j_{2+}p_{2+} Sol_{ét} \ar[u]^{(3.8)}_{∼} \ar[r]^{∼}_{(9.6)} & j_{2+} Sol_{ét} p_{2!} \ar[r]^{∼}_{(9.5)} & Sol_{ét} j_{2!}p_{2!} \ar[u]^{∼}}

(in which each isomorphism has been labelled by the result that gives rise to it) commutes.

Proof. We may form the commutative diagram

\xymatrix{Y \ar[rr]^{j_{1}} \ar[r]^{j_{1}×p_{2}} \ar[dr]^{p_{2}} & W_{1} ×_{X} W_{2} \ar[d] \ar[r] & W_{1} \ar[d]^{p_{1}} \\  & W_{2} \ar[r]^{j_{2}} & X,}

in which the square is cartesian, and the morphism j_{1}×p_{2} is both an open immersion and a proper morphism. Applying parts (ii) of propositions 9.5 and 9.6 respectively to the upper and left-hand triangles, we see that it suffices to verify the claimed commutativity in the case that the pair of factorizations

f = p_{1}j_{1} = j_{2}p_{2}

arises from a cartesian square. In this case the result follows from the constructions of propositions 9.5 and 9.6, together with part (ii) of theorem 4.4.1. □

(9.7.4) Proof of theorem 9.7.1. Note that if we have the isomorphism of part (i) for two morphisms f and g, proposition 3.8 will allows us to construct such an isomorphism for the composite fg. (In other words, we can take the diagram of part (ii) to define the isomorphism for the composite fg.) Thus if f is allowable, and f = f_{1} ⋯ f_{n} is some factorization of f into a composite of morphisms f_{i} between smooth k-schemes, such that each f_{i} is an open immersion or a proper morphism, we may construct a particular isomorphism Sol_{ét} f_{+} ⟶^{∼} f_{!} Sol_{ét} . (At least one such factorization exists, since we have assumed that f is allowable.) We will prove that the isomorphism so constructed is independent of choice of such a factorization of f. This will in particular provide a proof of part (ii).

We first suppose that f is a closed immersion equipped with a factorization f = f_{1} ⋯ f_{n} of f into a composite of open immersions and proper morphisms. Using induction on n, we will show that the isomorphism of part (i) obtained by using this factorization of f is the same as that obtained by applying proposition 9.6 directly to the closed immersion (and hence proper morphism) f itself.

If n = 1 this is a tautology, and if n = 2 it follows from theorem 9.7.3. The general case now follows immediately by induction on n, since f being a closed immersion shows that each of the partial composites f_{i} ⋯ f_{n} (1 ≤ i ≤ n) is also a closed immersion.

Now suppose that f : Y → X is an allowable morphism of smooth k-schemes, and that we have a factorization

\xymatrix{f : Y = Y_{n} \ar[r]^{f_{n}} & Y_{n−1} \ar[r]^{f_{n−1}} & ⋯ \ar[r]^{f_{2}} & Y_{1} \ar[r]^{f_{1}} & X,}

in which each of the morphisms f_{i} is either an open immersion or a proper morphism. We will show that the isomorphism induced by this factorization is independent of the given factorization. In order to do this, we choose a factorization of f,

\xymatrix{f : Y \ar[r]^{i} & U \ar[r]^{j} & W \ar[r]^{p} & X,}

in which i is a closed immersion, j is an open immersion, and p is a smooth proper morphism. We will prove that the isomorphism of part (i) obtained using the factorization f = f_{1} ⋯ f_{n} is the same as that obtained using the factorization f = pji. Since this latter factorization is independent of the factorization f = f_{1} ⋯ f_{n} with which we began, the independence will follow.

The argument depends on a consideration of the commutative diagram

[figure: Y_{n} ×_{X} U; f_{n}×id_{U}; ⋯; f_{2}×id_{U}; Y_{1} ×_{X} U; f_{1}×id_{W}; U; j; Y; id×(ji); id×i; i; Y_{n} ×_{X} W; f_{n}×id_{W}; ⋯; f_{2}×id_{W}; Y_{1} ×_{X} W; f_{1}×id_{W}; W; p; Y_{n}; f_{n}; ⋯; f_{2}; Y_{1}; f_{1}; X.]

(Because p is smooth, each of the k-schemes appearing in this diagram is smooth.)

The case of a closed immersion considered above shows that we get the same isomorphism whether we use the closed immersion i or the factorization

\xymatrix{Y \ar[r] & Y_{n} ×_{X} U \ar[r] & ⋯ \ar[r] & Y_{1} ×_{X} U \ar[r] & U.}

Also in each square, opposite sides are either both open immersions or both proper morphisms. Thus propositions 9.5 and 9.6, and lemma 9.7.3 show that we get the same isomorphism whichever way we go around each square. Finally, the edges of each of the two left-hand triangles are either open immersions or proper maps, so that proposition 9.6 and lemma 9.7.3 show that we get the same isomorphism whichever way we go around each of the triangles. This completes the proof of the first two parts of the theorem. The compatibility with inseparable base-change follows from remark 4.6, which observes that the constructions of that section, and hence those of propositions 9.5 and 9.6, and hence that of this theorem, are compatible with change of ground field. □

Proposition 9.8. Let X be a smooth k-scheme. Then the image of the functor Sol_{ét} lies in D_{c}^{b}(X_{ét}).

Proof. First note that the compatibility of Sol_{ét} with inseparable base-change allows us to replace k by its perfect closure, and so for the duration of the proof we may assume that k is perfect.

We proceed by induction on the dimension of the support of M^{•}. Let Z be this support. An argument using truncation and induction on the length of a complex shows that we can restrict our attention to a single locally finitely generated unit O_{F^{r},Xét}-module M supported on Z.

By proposition 6.7.5 we may find U a smooth dense open subscheme of Z (given its reduced induced structure) such that, if f denotes the immersion of U into X, then f^{!}M is a unit F^{r}-crystal on U. Proposition 5.10.1 shows that there is a morphism M → f_{+}f^{!}M whose cone is supported on Z \ U, which has dimension strictly less than that of Z. Thus to prove the proposition for M, it suffices to prove it for f_{+}f^{!}M.

Now Sol_{ét}(f_{+}f^{!}M) = f_{!} Sol_{ét}(f^{!}M), and f_{!} preserves the property of having constructible cohomology sheaves. Thus it suffices to show that Sol_{ét}(f^{!}M) has constructible cohomology sheaves. However, corollary 9.4.2 shows that

Sol_{ét}(f^{!}M) = \underline{RHom}_{O_{F^{r} ,Uét}}^{•} (f^{!}M, O_{Uét})[d_{U}] ⟶^{∼} L[d_{U}]

for some locally constant, and hence constructible, sheaf L on U. This proves the proposition. □

Finally we show that Sol_{ét} is compatible with tensor products:

Proposition 9.9. If M^{•}, N^{•} are in D_{lfgu}^{b}(O_{F^{r},Xét}) then there is a natural isomorphism in D_{c}^{b}(X_{ét}), which is compatible (in an obvious sense) with inseparable base-change.

Sol_{ét}(M^{•}) ⊗_{𝔽_{q}} Sol_{ét}(N^{•}) ⟶^{∼} Sol_{ét}(M^{•} ⊗^{𝕃}_{O_{X}} N^{•})[d_{X}].

Proof. If we combine lemmas 1.13.1 and 1.9.3 we obtain a natural transformation

Sol_{ét}(M^{•}) ⊗ Sol_{ét}(N^{•})
=\underline{RHom}_{O_{F^{r} ,X}}^{•} (M^{•}, O_{X})[d_{X}] ⊗_{𝔽_{q}} \underline{RHom}_{O_{F^{r} ,X}}^{•} (N^{•}, O_{X})[d_{X}]
→\underline{RHom}_{O_{F^{r} ,X}}^{•} (M^{•} ⊗^{𝕃}_{O_{X}} N^{•}, O_{X} ⊗^{𝕃}_{O_{X}} O_{X})[2d_{X}]
=\underline{RHom}_{O_{F^{r} ,X}}^{•} (M^{•} ⊗^{𝕃}_{O_{X}} N^{•}, O_{X})[2d_{X}]
= Sol_{ét}(M^{•} ⊗^{𝕃}_{O_{X}} N^{•})[d_{X}].

This is the natural transformation which we will show is an isomorphism. Lemma 1.13.1 together with the compatibility of Sol_{ét} with inseparable base-change shows that this natural transformation is compatible with inseparable base-change. In particular this allows us to replace k by its perfect closure, and so for the remainder of the proof we may assume that k is perfect. Suppose first that M^{•} = O_{X}. Then

M^{•} ⊗^{𝕃}_{O_{X}} N^{•} = O_{X} ⊗^{𝕃}_{O_{X}} N^{•} = N^{•}, and

Sol_{ét}(M^{•}) = Sol_{ét}(O_{X}) = 𝔽_{q} [d_{X}].

Thus the natural transformation constructed above reduces to the isomorphism

𝔽_{q} [d_{X}] ⊗_{𝔽_{q}} Sol_{ét}(N^{•}) ⟶^{∼} Sol_{ét}(N^{•})[d_{X}],

and we are done in this case. The case when M^{•} = O_{X}^{n} follows from this, and the case that M^{•} = M is a unit F^{r}-crystal likewise follows, since then M is étale locally isomorphic to O_{X}^{n}.

We now turn to the case of an arbitrary complex M^{•}, proceeding by induction on the dimension of the support of M^{•}. Let Z be this support. By truncation and induction on the length of M^{•}, we reduce to the case when M^{•} = M is a single locally finitely generated unit O_{F^{r},X}-module. Then proposition 6.7.5 shows that there is a smooth dense open subscheme U of Z (given its reduced induced structure), such that, if f denotes the immersion of U in X, then f^{!}M is a unit F^{r}-crystal on U. Proposition 5.10.1 now shows that there is a map M → f_{+}f^{!}M whose support is contained in Z \ U, which is of dimension strictly less than that of Z. By our inductive assumption, we are reduced to verifying that the natural transformation

(9.9.1)  Sol_{ét}(f_{+}f^{!}M) ⊗_{𝔽_{q}} Sol_{ét}(N^{•}) → Sol_{ét}(f_{+}f^{!}M ⊗^{𝕃}_{O_{X}} N^{•})[d_{X}]

is an isomorphism.

Now proposition 4.2 provides a natural isomorphism

f_{+}f^{!}M ⊗^{𝕃}_{O_{X}} N^{•} ⟶^{∼} f_{+}(f^{!}M ⊗^{𝕃}_{O_{U}} f^{!}N^{•})[d_{X/U}].

Thus we obtain the following commutative diagram:

\xymatrix{Sol_{ét}(f_{+}f^{!}M) ⊗_{𝔽_{q}} Sol_{ét}(N^{•}) \ar[r]_{(1)}^{∼} \ar[d]^{(9.9.1)} & f_{!} Sol_{ét}(f^{!}M) ⊗_{𝔽_{q}} Sol_{ét}(N^{•}) \ar[d]^{∼} \\ Sol_{ét}(f_{+}f^{!}M ⊗^{𝕃}_{O_{X}} N^{•})[d_{X}] \ar[d]_{(4.2)}^{∼} & f_{!}(Sol_{ét}(f^{!}M) ⊗_{𝔽_{q}} f^{−1} Sol_{ét}(N^{•})) \ar[d]_{(3)}^{∼} \\ Sol_{ét}(f_{+}(f^{!}M ⊗^{𝕃}_{O_{U}} f^{!}N^{•})[d_{X/U}])[d_{X}] \ar[d]^{∼} & f_{!}(Sol_{ét}(f^{!}M) ⊗_{𝔽_{q}} Sol_{ét}(f^{!}N^{•})) \ar[d]_{(4)}^{∼} \\ Sol_{ét}(f_{+}(f^{!}M ⊗^{𝕃}_{O_{U}} f^{!}N^{•}))[d_{U}] \ar[r]_{(2)}^{∼} & f_{!}(Sol_{ét}(f^{!}M ⊗^{𝕃}_{O_{X}} f^{!}N^{•}))[d_{U}].}

Here (1) and (2) are deduced from the natural isomorphism Sol_{ét} f_{+} ⟶^{∼} f_{!} Sol_{ét}, isomorphism (3) is deduced from the natural isomorphism Sol_{ét} f^{!} ⟶^{∼} f^{−1} Sol_{ét}, and isomorphism (4) is deduced from that case of the proposition which we proved above, since f^{!}M is a unit F^{r}-crystal on U.

Thus we see that the natural transformation (9.9.1) is indeed an isomorphism, and we are done. □

10. the functor M_{ét}

(10.1) Suppose that X is a k-scheme. Then the sheaf of rings O_{F^{r},Xét} is flat over 𝔽_{q}, and so any injective O_{F^{r},Xét}-module is also an injective 𝔽_{q}-module. (The forgetful functor “regard an O_{F^{r},Xét}-module as a 𝔽_{q}-module” is right adjoint to the exact functor O_{F^{r},Xét} ⊗_{𝔽_{q}} –.) Thus if M^{•} is any complex in D^{b}(O_{F^{r},X}), any resolution of M^{•} by injective O_{F^{r},Xét}-modules is also an injective resolution of M^{•} by injective 𝔽_{q}-sheaves. Thus we regard \underline{RHom}_{𝔽_{q}}^{•}(–, M^{•}) as a contravariant functor

\underline{RHom}_{𝔽_{q}}^{•}(–, M^{•}) : D_{c}^{b}(X_{ét}) → D^{+}(O_{F^{r},Xét}).

Definition 10.1.1. If X is a smooth k scheme, we define the functor

M_{ét} : D_{c}^{b}(X_{ét}) → D^{+}(O_{F^{r},Xét})

as follows:

M_{ét}(F^{•}) = \underline{RHom}_{𝔽_{q}}^{•}(F^{•}, O_{Xét})[d_{X}].

Lemma 10.1.2. There is a commutative diagram of natural transformations

[figure: F_{X}^{r∗} M_{ét}(F^{•}); ϕ_{Mét(F^{•})}; F_{X}^{r∗}\underline{RHom}_{𝔽_{q}}^{•}(F^{•}, O_{Xét})[d_{X}]; \underline{RHom}_{𝔽_{q}}^{•}(F^{•}, F_{X}^{r∗}O_{Xét}); \underline{RHom}^{•}(F^{•},ϕ_{O_{Xét}} ); M_{ét}(F^{•}); \underline{RHom}_{𝔽_{q}}^{•}(F^{•}, O_{Xét})[d_{X}]]

in which the upper horizontal arrow is an isomorphism. Since ϕ_{O_{Xét}} is an isomorphism, we deduce that the same is true of ϕ_{Mét(F^{•})}.

Proof. Let I^{•} be a right resolution of O_{Xét} by left O_{F^{r},X}-modules, which are injective as 𝔽_{q}-modules. The sheaf O_{Xét}^{(r)} is locally free as a right O_{Xét}-module, and so F_{X}^{r∗}I^{•} is a direct limit of injective 𝔽_{q}-sheaves, and so (by the usual noetherian argument, since any 𝔽_{q} sheaf is the direct limit of its constructible sub-modules) we see that F_{X}^{r∗}I^{•} is again a complex of injective 𝔽_{q}-modules.

Thus the commutative diagram

\xymatrix{F_{X}^{r∗}\underline{Hom}_{𝔽_{q}}^{•}(F^{•}, I^{•})[d_{X}] \ar[d] \ar[r] & \underline{Hom}_{𝔽_{q}}^{•}(F^{•}, F_{X}^{r∗}I_{Xét}^{•}) \ar[dl]^{\underline{Hom}^{•}(F^{•},ϕ_{I{• \atop Xét}} )} \\ \underline{Hom}_{𝔽_{q}}^{•}(F^{•}, I_{Xét}^{•})[d_{X}]}

is a realization of the diagram of the lemma on the level of complexes.

To show that the horizontal arrow is an isomorphism, we may use truncation and induction on the length of the complex F^{•} to reduce to the case in which it is a single constructible sheaf F. Since F is noetherian, we may pass the tensor product with the locally free sheaf O_{Xét}^{(r)} through \underline{Hom}_{𝔽_{q}}(F, –). This completes the proof. □

Remark 10.1.3 We see from this lemma (via (2.9.1) and the remark following it) that the structural morphisms of the cohomology sheaves of M_{ét}(F^{•}) are isomorphisms. We will see in proposition 10.4 below that in fact M_{ét}(F^{•}) lies in D_{lfgu}^{b}(O_{F^{r},Xét}).

Remark 10.1.4 Let k′ be a purely inseparable algebraic extension of k, and let X be a smooth k-scheme. If I^{•} is a resolution of O_{Xét} by injective O_{F^{r},Xét}-modules, then k′ ⊗_{k} I^{•} is a resolution of O_{Xét^{′}} by O_{F^{r},Xét^{′}}-modules, and so we may find a quasi-isomorphism k′ ⊗_{k} I^{•} → I^{′•}, where I^{′•} is a resolution of O_{Xét^{′}} by injective O_{F^{r},Xét^{′}}-modules. Thus for any object F^{•} of D_{c}^{b}(X_{ét}) there is a natural morphism

k′ ⊗_{k} \underline{Hom}_{𝔽_{q}}^{•}(F^{•}, I^{•}) → \underline{Hom}_{𝔽_{q}}^{•}(F^{•}, k′ ⊗_{k} I^{•}) → \underline{Hom}_{𝔽_{q}}^{•}(F^{•}, I^{′•}),

giving rise to the commutative diagram

\xymatrix{D_{c}^{b}(X_{ét}) \ar[r]^{∼} \ar[d]^{Mét} & D_{c}^{b}(X′_{ét}) \ar[d]^{Mét} \\ D^{+}(O_{F^{r},Xét}) \ar[r]^{k′⊗_{k}–} & D^{+}(O_{F^{r},Xét^{′}}).}

This diagram expresses the sense in which the functor M_{ét} is compatible with inseparable base-change.

(10.2) The following simple lemma will be the basis for all our computations of M_{ét}:

Lemma 10.2.1. If X is a smooth k-scheme, then M_{ét}(𝔽_{q}) = O_{Xét}[d_{X}].

Proof. This follows immediately from the definition. □

(10.3) In this section we will prove the existence of a natural isomorphism M_{ét} f_{!} = f_{+} M_{ét} when f : Y → X is an immersion of smooth k-schemes. Note that the functor f_{+} is defined on the category D^{+}(O_{F^{r},Yét}), by remark 3.5.15.

Lemma 10.3.1. If f : Y → X is an open immersion of smooth k-schemes, then there is a natural isomorphism of functors:

M_{ét} f_{!} ⟶^{∼} f_{+} M_{ét} .

Proof. Let F^{•} be an object in D_{c}^{b}(X_{ét}), and let I^{•} be a resolution of O_{Xét} by injective O_{F^{r},Xét}-modules (which are also 𝔽_{q}-injective, by the above discussion). Then f^{−1}I^{•} is an injective resolution of O_{Yét} in the category of O_{F^{r},Yét}-modules (since f^{−1} is right adjoint to the exact functor f_{!}).

We compute

M_{ét}(f_{!}F^{•}) = \underline{RHom}_{𝔽_{q}}^{•}(f_{!}F^{•}, O_{Xét})[d_{X}]

⟶^{∼} \underline{Hom}_{𝔽_{q}}^{•}(f_{!}F^{•}, I^{•})[d_{X}] ⟶^{∼} f_{∗}\underline{Hom}_{𝔽_{q}}^{•}(F^{•}, f^{−1}I^{•})[d_{X}]

(by the adjointness of f_{!} and f^{−1}). Now for any F^{i}, \underline{Hom}_{𝔽_{q}}^{•}(F^{i}, –) is right adjoint to the exact functor – ⊗_{𝔽_{q}} F^{i} on the category of 𝔽_{q} sheaves, and so takes 𝔽_{q}-injectives to 𝔽_{q}-injectives. Thus \underline{Hom}_{𝔽_{q}}^{•}(F^{•}, f^{−1}I^{•})[d_{X}] is a complex of 𝔽_{q}-injectives, and

f_{∗}\underline{Hom}_{𝔽_{q}}^{•}(F^{•}, f^{−1}I^{•})[d_{X}] = Rf_{∗}\underline{Hom}_{𝔽_{q}}^{•}(F^{•}, f^{−1}I^{•})[d_{X}].

Also, X and Y have the same dimension, since f is an open immersion, and so

Rf_{∗}\underline{Hom}_{𝔽_{q}}^{•}(F^{•}, f^{−1}I^{•})[d_{X}] ⟶^{∼} Rf_{∗}\underline{RHom}_{𝔽_{q}}^{•}(F^{•}, O_{Yét})[d_{Y}] ⟶^{∼} Rf_{∗} M_{ét}(F^{•}).

Now we saw in lemma 4.3.1 that

f_{+} M_{ét}(F^{•}) ⟶^{∼} Rf_{∗} M_{ét}(F^{•}).

Putting all these natural isomorphisms together, we see that

M_{ét}(f_{!}F^{•}) ⟶^{∼} f_{+} M_{ét}(F^{•}),

proving the lemma. □

Proposition 10.3.2. If f : Y → X is a closed immersion of smooth k-schemes, there is a natural isomorphism

f_{+} M_{ét} ⟶^{∼} M_{ét} f_{!}.

Proof. We begin by describing the natural transformation. Recall by example 5.9.6 that we have canonical morphisms

f_{+}O_{Yét}[d_{Y/X}] ⟶^{∼} f_{+}f^{!}O_{Xét} ⟶^{∼} RΓ_{Y} (O_{Xét}) → O_{Xét},

whose composite is the trace map of proposition 4.4.9. Also, for the closed immersion f_{∗}, no derived functors are required to define the functor f_{+} (by remark 3.5.15 and the exactness of f_{∗}); it is given by the simple formula

f_{+}(–) = f_{∗}(O_{F^{r},Xét←Yét} ⊗_{O_{F^{r} ,Yét}} –).

Thus for any complex F^{•} in D_{c}^{b}(Y_{ét}) we have the natural transformation

f_{+} M_{ét}(F^{•}) ⟶^{∼} f_{∗}(O_{F^{r},Xét←Yét} ⊗_{O_{F^{r} ,Yét}} \underline{RHom}_{𝔽_{q}}^{•}(F^{•}, O_{Yét})[d_{Y}])

⟶ \underline{RHom}_{𝔽_{q}}^{•}(f_{∗}F^{•}, f_{∗}(O_{F^{r},Xét←Yét} ⊗_{O_{F^{r} ,Yét}} O_{Yét})[d_{Y}])

= \underline{RHom}_{𝔽_{q}}^{•}(f_{∗}F^{•}, f_{+}O_{Yét}[d_{Y/X}])[d_{X}]

⟶ \underline{RHom}_{𝔽_{q}}^{•}(f_{∗}F^{•}, O_{Xét})[d_{X}]

⟶^{∼} M_{ét}(f_{∗}F^{•}),

which we will show is an isomorphism. This can be checked locally on X_{ét}. Note also that both sides of this homomorphism vanish on the complement of Y in X, so it suffices to verify that we obtain an isomorphism in a neighbourhood in X_{ét} of each point of Y. As the last step of our initial reductions, note that the compatibility of M_{ét} with inseparable base-change allows us to replace k by its perfect closure, so that for the duration of the proof we may assume that k is perfect. We now proceed by cases.

(i) Local systems: We be begin by considering the case F^{•} = 𝔽_{q} in degree zero. Then the natural transformation reduces reduces to composite of the isomorphisms

f_{+}O_{Yét}[d_{Y}] ⟶^{∼} RΓ_{Y} (O_{X})[d_{X}] ⟶^{∼} \underline{RHom}_{𝔽_{q}}^{•}(f_{∗}𝔽_{q}, O_{Xét})[d_{X}].

The result now follows for F^{•} = L, a local system placed in degree zero, since L is isomorphic to a direct sum of copies of 𝔽_{q} étale locally on Y.

(ii) Constructible sheaves: Suppose that F is a constructible sheaf of 𝔽_{q}-vector spaces on Y_{ét}. We will prove that the natural map f_{+} M_{ét}(F) → M_{ét}(f_{!}F) is an isomorphism by induction on the dimension of the support of F.

Thus, let Z be the support of F. There is a dense open subscheme V of Z (given its reduced induced structure) which is smooth over 𝔽_{q}, such that over V the sheaf F restricts to a local system L of finite rank 𝔽_{q}-vector spaces. Let g be the immersion of V into Y. Let W = Z \ V, a closed subset of X of dimension less than that of Z, and let h : W → Y be the closed immersion. We have a short exact sequence

0 → g_{!}L = g_{!}g^{−1}F → F → h_{!}h^{−1}F → 0.

By induction, we already know the result for h_{!}h^{−1}F, so it suffices to prove it for g_{!}L.

We may factor the immersion g as the composite of an open immersion j with a closed immersion i: g = ji. We also factor the immersion fj as the composite of an an open immersion l and a closed immersion k: fj = lk. We obtain a diagram

[figure: f_{+} M_{ét}(g_{!}L); f_{+} M_{ét}(j_{!}i_{!}L); ∼; f_{+}j_{+} M_{ét}(i_{!}L); l_{+}k_{+} M_{ét}(i_{!}L); ∼; M_{ét}(f_{!}g_{!}L); M_{ét}(l_{!}k_{!}i_{!}L); ∼; l_{+} M_{ét}(k_{!}i_{!}L); l_{+}k_{+}i_{+} M_{ét}(L); ∼]

in which the two right pointing horizontal arrows are the natural isomorphisms of lemma 10.3.1 applied to the open immersions j and l, the left pointing horizontal arrow is the isomorphism of part (i) applied to the closed immersion ki, and the right vertical arrow is the inverse of the isomorphism of part (i) applied to the closed immersion i. One easily verifies that this diagram commutes, and thus that the left vertical arrow is also an isomorphism, as required.

We can now complete the proof of the proposition. Indeed, it follows from part (ii) by truncation and induction on the length of a complex in D_{c}^{b}(Y_{ét}). □

Corollary 10.3.3. If f : Y → X is an immersion of smooth k-schemes, then there is a natural isomorphism

f_{+} M_{ét} ⟶^{∼} M_{ét} f_{!}.

Proof. Since any immersion decomposes as the composition of an open and closed immersion, this follows from the preceding two results. □

Proposition 10.4. If X is a smooth k-scheme, then for any complex F^{•} contained in D_{c}^{b}(X_{ét}), the complex M_{ét}(F^{•}) of O_{F^{r},Xét}-modules has only finitely many nonvanishing cohomology sheaves, and these are locally finitely generated unit O_{F^{r},Xét}- modules. Thus we may regard M_{ét} as a functor

M_{ét} : D_{c}^{b}(X_{ét}) → D_{lfgu}^{b}(O_{F^{r},Xét})).

Proof. The compatibility of M_{ét} with inseparable base-change allows us to replace k by its perfect closure, and so for the duration of the proof we may assume that k is perfect. Let F^{•} be a complex in D_{c}^{b}(X_{ét}); by truncation and induction on the length of a complex it suffices to treat the case when F^{•} = F is a single constructible sheaf in degree zero. We proceed by induction on the dimension of the support of F.

Let Z be the support of F. There is a dense open subscheme Y of Z (given its reduced induced structure) which is smooth as a k-scheme, and over which F restricts to a local system L of finite rank 𝔽_{q}-vector spaces.

Let f : Y → X denote the immersion of Y into X. Let W = Z \ Y be the complement of Y in Z, a closed subset of X of dimension less than that of Z. Let i : W → X denote the closed immersion of W into X.

There is a short exact sequence

0 → f_{!}L = f_{!}f^{−1}F → F → i_{!}i^{−1}F → 0.

Applying M_{ét} yields a distinguished triangle

M_{ét}(i_{!}i^{−1}F) → M_{ét}(F) → M_{ét}(f_{!}L) → M_{ét}(i_{!}i^{−1}F)[1].

Since i_{!}i^{−1}F is supported on W, it follows by induction that M_{ét}(i_{!}i^{−1}F) has only finitely many non-vanishing cohomology sheaves, and that these are locally finitely generated unit O_{F^{r},Xét}-modules.

Thus to conclude the corresponding result for M_{ét}(F), it suffices to prove it for M_{ét}(f_{!}L). We have seen that M_{ét}(f_{!}L) ⟶^{∼} f_{+} M_{ét}(L). Since f_{+} : D_{lfgu}^{b}(O_{F^{r},Yét}) → D_{lfgu}^{b}(O_{F^{r},Xét}), it is enough to show that M_{ét}(L) is a complex in D_{lfgu}^{b}(O_{F^{r},Xét}). Now M_{ét}(L) = \underline{RHom}_{𝔽_{q}}^{•}(L, O_{Yét})[d_{Y}]. Locally on Y_{ét}, L is isomorphic to (𝔽_{q})^{n} for some n, and so locally on Y_{ét}, the O_{F^{r},Yét}-module \underline{RHom}_{𝔽_{q}}^{•}(L, O_{Yét}) is isomorphic to O_{Yét}^{n}, which is a finitely generated unit O_{F^{r},Yét}-module. □

11. riemann-hilbert correspondence

(11.1) We now have all the machinery necessary to prove our Riemann-Hilbert correspondence between constructible 𝔽_{q}-sheaves on X_{ét} and locally finitely generated unit O_{F^{r},Xét}-modules. We begin by noting that there are natural transformations η : id → Sol_{ét} M_{ét} and ψ : id → M_{ét} Sol_{ét}. If I^{•} is a right resolution of O_{Xét} by injective left O_{F^{r},Xét}-modules, then for any complex M^{•} in D_{lfgu}^{b}(O_{F^{r},Xét}) we define ζ_{M^{•}} as the composite morphism

M^{•} → \underline{Hom}_{𝔽_{q}}^{•}(\underline{Hom}_{O_{F^{r} ,Xét}}^{•} (M^{•}, I^{•}), I^{•})

→ \underline{Hom}_{𝔽_{q}}^{•}(\underline{Hom}_{O_{F^{r} ,Xét}}^{•} (M^{•}, I^{•})[d_{X}], I^{•})[d_{X}].

Similarly, if F^{•} is a complex in D_{c}^{b}(X_{ét}), we define η_{F^{•}} as the composite morphism

F^{•} → \underline{Hom}_{O_{F^{r} ,Xét}}^{•} (\underline{Hom}_{𝔽_{q}}^{•}(F^{•}, O_{Xét}), O_{Xét})

→ \underline{Hom}_{O_{F^{r} ,Xét}}^{•} (\underline{Hom}_{𝔽_{q}}^{•}(F^{•}, O_{Xét})]d_{X}], O_{Xét})[d_{X}].

(We refer to [Con, I §1] for a discussion of the sign conventions involved in the definitions of these morphisms.) It follows from the analysis of signs in [Con, I §1] that the composites

M_{ét}(F^{•}) ⟶^{ζ_{Mét(F^{•})}} M_{ét}(Sol_{ét}(M_{ét}(F^{•}))) ⟶^{Mét(η_{F^{•}})} M_{ét}(F^{•})

and

Sol_{ét}(M^{•}) ⟶^{η_{Solét(M^{•})}} Sol_{ét}(M_{ét}(Sol_{ét}(M^{•}))) ⟶^{Solét(ζ_{M^{•}})} Sol_{ét}(M^{•})

are the identity morphisms. Also, it is immediate that both ζ and η are compatible with base-change by an inseparable field extension, in an obvious sense.

We will prove that both ζ and η are isomorphisms, and consequently will conclude that Sol_{ét} and M_{ét} induce an equivalence of categories between D_{lfgu}^{b}(O_{F^{r},Xét}) and D_{c}^{b}(X_{ét}). The proof will be via an excision argument, combining theorem 9.7.1 and corollary 10.3.3. Before making this argument, we must verify that these two results are compatible with the natural transformations ζ and η in an appropriate sense.

Proposition 11.2. If f : Y → X is an immersion of smooth k-schemes and if M^{•} is an object of D_{lfgu}^{b}(O_{F^{r},Xét}), then the diagram of natural isomorphisms

\xymatrix{f_{+}M^{•} \ar[d]^{f_{+}(ζ_{M^{•}})} \ar[r]^{ζ_{f_{+}M^{•}}} & M_{ét}(Sol_{ét}(f_{+}M^{•})) \ar[d]^{∼}_{(9.7.1)} \\ f_{+} M_{ét}(Sol_{ét}(M^{•})) \ar[r]^{∼}_{(10.3.3)} & M_{ét}(f_{!} Sol_{ét}(M^{•}))}

(in which each natural isomorphism has been labelled by the result that gives rise to it) commutes.

Similarly, if F^{•} is an object of D_{c}^{b}(X_{ét}), then the diagram of natural isomorphisms

\xymatrix{f_{!}F^{•} \ar[d]^{f_{!}(η_{F^{•}})} \ar[r]^{η_{f_{!}F^{•}}} & Sol_{ét}(M_{ét}(f_{!}F^{•})) \ar[d]^{∼}_{(10.3.3)} \\ f_{!} Sol_{ét}(M_{ét}(F^{•})) \ar[r]^{∼}_{(9.7.1)} & Sol_{ét}(f_{+} M_{ét}(M^{•}))}

(in which each natural isomorphism has been labelled by the result that gives rise to it) commutes.

Proof. The natural transformations of both theorem 9.7.1 and of corollary 10.3.3 are defined by factoring f as a closed immersion followed by an open immersion. One verifies that the diagrams whose commutativity is to be checked are compatible with compositions in an obvious sense, and thus it suffices to verify that each diagram commutes in the case when f is either an open or a closed immersion.

If f is an open immersion, then the commutativity of both diagrams may be verified after restricting to Y, at which point it is immediate from the constructions.

If f is a closed immersion we have the diagram

\xymatrix{f_{+}M^{•} \ar[dd] \ar[dr] \\  & \underline{RHom}^{•}(\underline{RHom}^{•}(f_{+}M^{•}, O_{Xét})[d_{X}], O_{Xét})[d_{X}] \ar[dd] \\ \underline{RHom}^{•}(\underline{RHom}^{•}(f_{+}M^{•}, f_{+}O_{Yét}[d_{Y}]), f_{+}O_{Yét}[d_{Y}]) \ar[dr] \ar[dd] \\  & \underline{RHom}^{•}(\underline{RHom}^{•}(f_{+}M^{•}, f_{+}O_{Yét}[d_{Y}]), O_{Xét})[d_{X}] \ar[dd] \\ \underline{RHom}^{•}(f_{∗}\underline{RHom}^{•}(M^{•}, O_{Yét})[d_{Y}], f_{+}O_{Yét})[d_{Y}] \ar[dr] \\  & \underline{RHom}^{•}(f_{∗}\underline{RHom}^{•}(M^{•}, O_{Yét})[d_{Y}], O_{Xét})[d_{X}],}

as well as the diagram

\xymatrix{ & f_{+}M^{•} \ar[d] \ar[ddl] \\  & \underline{RHom}^{•}(\underline{RHom}^{•}(f_{+}M^{•}, f_{+}O_{Yét}[d_{Y}]), f_{+}O_{Yét}[d_{Y}]) \ar[dd] \\ f_{+}\underline{RHom}^{•}(\underline{RHom}^{•}(M^{•}, O_{Yét})[d_{Y}], O_{Yét})[d_{Y}] \ar[dr] \\  & \underline{RHom}^{•}(f_{∗}\underline{RHom}^{•}(M^{•}, O_{Yét})[d_{Y}], f_{+}O_{Yét})[d_{Y}].}

The first of these obviously commutes, while the second is also seen to commute once one unwinds the “double duality” maps that it involves. If one glues these two diagrams along their common edge one obtains (an expanded version of) the first diagram in the statement of the proposition, and so one sees that this diagram commutes.

One sees that the second diagram in the statement of the lemma arising from the closed immersion f commutes by considering an analogous pair of diagrams, whose construction we leave to the reader. □

Theorem 11.3. The natural transformations ζ : id → M_{ét} Sol_{ét} and η : id → Sol_{ét} M_{ét} are isomorphisms. Thus the functor M_{ét} : D_{c}^{b}(X_{ét}) → D_{lfgu}^{b}(O_{F^{r},Xét}) is an anti-equivalence of triangulated categories, with Sol_{ét} : D_{lfgu}^{b}(O_{F^{r},Xét}) → D_{c}^{b}(X_{ét}) providing a quasi-inverse.

Suppose that F^{•} and G^{•} are two complexes in D_{c}^{b}(X_{ét}), and that M^{•} and N^{•} are two complexes in D_{lfgu}^{b}(O_{F^{r},Xét}). Then there are natural isomorphisms of complexes in D^{b}(X_{ét}),

(11.3.1)  RHom^{•}_{𝔽_{q}}(F^{•}, G^{•}) ⟶^{∼} RHom^{•}_{O_{F^{r} ,Xét}} (M_{ét}(G^{•}), M_{ét}(F^{•}))

and

(11.3.2)  RHom^{•}_{O_{F^{r} ,Xét}} (M^{•}, N^{•}) ⟶^{∼} RHom^{•}_{𝔽_{q}}(Sol_{ét}(N^{•}), Sol_{ét}(M^{•})),

a natural isomorphism of complexes in D_{lfgu}^{b}(O_{F^{r},Xét}),

(11.3.3)  M_{ét}(F^{•}) ⊗^{𝕃}_{O_{Xét}} M_{ét}(G^{•}) ⟶^{∼} M_{ét}(F^{•} ⊗_{𝔽_{q}} G^{•})[d_{X}],

and a natural isomorphism of complexes in D_{c}^{b}(X_{ét}),

(11.3.4)  Sol_{ét}(M^{•}) ⊗_{𝔽_{q}} Sol_{ét}(N^{•}) ⟶^{∼} Sol_{ét}(M^{•} ⊗^{𝕃}_{O_{Xét}} N^{•})[d_{X}].

If f : Y ⟶ X is any map of smooth k-schemes, then there is a natural isomorphism of complexes in D_{lfgu}^{b}(O_{F^{r},Yét}),

(11.3.6)  f^{!} M_{ét}(F^{•}) ⟶^{∼} M_{ét}(f^{−1}F^{•}),

and a natural isomorphism of complexes in D_{c}^{b}(Y_{ét}),

(11.3.6)  Sol_{ét} f^{!}(M^{•}) ⟶^{∼} f^{−1} Sol_{ét}(M^{•}).

If furthermore f is allowable (in the sense of (9.7)) then there is a natural isomorphism of complexes in D_{lfgu}^{b}(O_{F^{r},Xét}),

(11.3.7)  M_{ét} f_{!}(F^{•}) ⟶^{∼} f_{+} M_{ét}(F^{•}),

and a natural isomorphism of complexes in D_{c}^{b}(X_{ét}),

(11.3.8)  Sol_{ét} f_{+}(M^{•}) ⟶^{∼} f_{!} Sol_{ét}(M^{•}).

Finally, this equivalence of categories and each of these natural isomorphisms is compatible in an obvious sense with inseparable base-change.

Proof. We begin by proving that ζ and η are natural isomorphisms. It has already been observed that these morphisms are compatible with inseparable base-change; in particular, we may replace k by its perfect closure, and so assume that k is perfect.

Let us suppose first that F^{•} is a complex in D_{c}^{b}(X_{ét}), and show that η : F^{•} → Sol_{ét}(M_{ét}(F^{•})) is an isomorphism. An argument using truncation and induction on the length of a complex shows that it suffices to prove this for a single constructible sheaf of 𝔽_{q}-vector spaces F.

We will argue by induction on the dimension of the support of F. Let Z be the support of F, and let Y be a dense open subset of Z which is a smooth k-scheme and over which F restricts to a local system L. Let W = Z \ Y be the complement of Y in Z; then W is a closed subset of X of dimension less than that of Z. Let i : W → X be the closed immersion of W into X, and f : Y → X the immersion of Y into X.

There is a short exact sequence

0 → f_{!}L = f_{!}f^{−1}F → F → i_{!}i^{−1}F → 0.

By induction, we already know the result for i_{!}i^{−1}F, so it suffices to prove it for f_{!}f^{−1}F. But by 9.7 and 10.3.3 we have

f_{!} Sol_{ét}(M_{ét}(L)) ⟶^{∼} Sol_{ét}(f_{+} M_{ét}(L)) ⟶^{∼} Sol_{ét}(M_{ét}(f_{!}L))

and proposition 11.2.1 shows that the morphism η_{f_{!}L} : f_{!}L → Sol_{ét}(M_{ét}(f_{!}L)) is obtained by composing this natural isomorphism with the morphism obtained by applying f_{!} to the morphism η_{L} : L → Sol_{ét}(M_{ét}(L)). Hence it suffices to show that this latter morphism is an isomorphism. This may be checked locally, and so we reduce to showing that 𝔽_{q} → Sol_{ét}(M_{ét}(𝔽_{q})) is an isomorphism. Now M_{ét}(𝔽_{q}) = O_{Xét}[d_{X}], by lemma 10.2.1, and the claim follows by example 9.3.1.

We have proved that η is an isomorphism. The proof that ζ is an isomorphism is formally similar. By the usual arguments we may reduce to proving it for a single module in degree zero. Consider then a locally finitely generated unit O_{F^{r},X}- module M. We proceed by induction on the dimension of the support Z of M. By proposition 6.7.5, there is a dense open subscheme U of Z (given its reduced induced structure), smooth as a k-scheme, such that if f : U → X denotes the immersion then f^{!}M is a unit F^{r}-crystal. As above, an excision argument reduces us to the case M = f_{+}N , where N is a unit F^{r}-crystal on U. Now proposition 11.2.1 shows that ζ_{f_{+}N} is an isomorphism provided ζ_{N} is. Thus we may assume that M = N and U = X. By lemma 9.4.1, N is étale locally isomorphic to O_{X}^{n}. As checking that ζ is an isomorphism is étale local, we are reduced to the case N = O_{X}, when the result is clear by example 9.3.1 and lemma 10.2.1, as above.

Now that we have established that M_{ét} and Sol_{ét} provide the asserted equivalence of categories, we will construct the natural isomorphisms (11.3.1) – (11.3.8). The isomorphisms (11.3.4), (11.3.6) and (11.3.8) were constructed in propositions 9.9, 9.3 and 9.7. We define (11.3.3), (11.3.5) and (11.3.7) to be the corresponding isomorphisms that arise from the equivalence of categories. (If f is an immersion then corollary 10.3.3 yields an alternative definition of (11.3.7), but this is shown by proposition 11.2.1 to agree with the more general definition just given.) In each of the results cited, the compatibility with inseparable base-change has been noted.

It remains to explain the isomorphism (11.3.1) and (11.3.2). This requires a slight digression on some sign conventions that are not treated in [Con, I §1].

(11.3.9) Suppose that A^{•}, B^{•} and C^{•} are three complexes of objects of an additive category. Then there is a natural morphism

ψ : Hom^{•}(A^{•}, B^{•}) → Hom^{•}(Hom^{•}(B^{•}, C^{•}), (A^{•}, C^{•}))

defined as follows: if f = (f^{q}) ∈ ∏_{q} Hom(A^{q}, B^{n+q}) is an element in the degree n member of the left-hand side, then

ψ(f) ∈ ∏_{q} Hom(∏_{p} Hom(B^{p}, C^{p+q}), ∏_{p} Hom(A^{p}, C^{n+p+q}))

(which is the degree n member of the right hand side) is defined via the formula

ψ(f)^{q} : (ϕ^{p}) → ((−1)^{nq}ϕ^{n+p}f^{p}).

The sign is chosen so that (when one uses the sign conventions of [Con, I §1] for the Hom^{•} complexes) ψ is a map of complexes. One checks that ψ is compatible in an obvious sense with the “double duality” maps

A^{•} → Hom^{•}(Hom^{•}(A^{•}, C^{•}))

and

B^{•} → Hom^{•}(Hom^{•}(B^{•}, C^{•})).

Also, it is compatible with making a simultaneous translation in A^{•} and B^{•}. The morphism induced by ψ on the degree n cohomology objects of each side is (up to a sign) the natural morphism

Hom(A^{•}, B^{•}[n]) → Hom(Hom^{•}(B^{•}[n], C^{•}), Hom^{•}(A^{•}, C^{•})).

(11.3.10) Construction of (11.3.1) and (11.3.2). Now let F^{•} and G^{•} be two complexes in D_{c}^{b}(X_{ét}), and let I^{•} be a right resolution of O_{X} by left O_{F^{r},Xét}- modules. Then we define the morphism (11.3.1) to be the derived category avatar of the composite

Hom_{𝔽_{q}}^{•}(F^{•}, G^{•}) ⟶ Hom_{O_{F^{r} ,Xét}}^{•} (\underline{Hom}_{𝔽_{q}}^{•}(G^{•}, I^{•}), \underline{Hom}_{𝔽_{q}}^{•}(F^{•}, I^{•}))

⟶^{∼} Hom_{O_{F^{r} ,Xét}}^{•} (\underline{Hom}_{𝔽_{q}}^{•}(G^{•}, I^{•})[d_{X}], \underline{Hom}_{𝔽_{q}}^{•}(F^{•}, I^{•})[d_{X}]),

where the first morphism is defined via the discussion of (11.3.9). Note that since the members of F^{•} are (trivially, since 𝔽_{q} is a field) 𝔽_{q}-flat, \underline{Hom}_{O_{F^{r} ,Xét}}^{•} (F^{•}, I^{•}) is a complex of injective left O_{F^{r},Xét}-modules. Thus this does realize on the level of complexes a morphism

RHom^{•}_{𝔽^{q}}(F^{•}, G^{•}) → RHom^{•}_{O_{F^{r} ,Xét}} (M_{ét}(G^{•}), M_{ét}(F^{•}))

between objects of D^{+}(X_{ét}). To see that this is an isomorphism, it suffices to consider the resulting map on cohomology modules; since M_{ét} is an equivalence of categories, these maps are indeed isomorphisms. Note that the right hand side of (11.3.1) is bounded, by proposition 5.12, and so we see now that the left hand side is also.

The morphism (11.3.2) can be defined in a similar way, or simply by applying the equivalence of categories to (11.3.1). Both definitions yield the same morphism, since the construction of (11.3.9) is compatible with double duality and translations. Finally, it is immediate that these constructions are compatible with inseparable base-change. □

(11.4) Thus far we have fixed q = p^{r}. Now we will study how the functors Sol_{ét} and M_{ét} behave when we change q. Thus in this section we denote Sol_{ét} and M_{ét} by Sol_{ét}_{,q} and M_{ét}_{,q} respectively.

(11.4.1) Suppose that 𝔽_{q} ⊂ 𝔽_{q′} ⊂ k, and write q = p^{r}, q′ = p^{r′}, so that r divides r′. If M^{•} is in D^{b}(O_{F^{r},X}) (respectively D^{b}(O_{F^{r},Xét})), then we denote by Res_{q}^{q′}M^{•} the complex M^{•} regarded as an object of D^{b}(O_{F^{r′} ,X}) (respectively D^{b}(O_{F^{r′} ,Xét})). In particular this definition makes sense for M^{•} a complex in D_{lfgu}^{b}(O_{F^{r},X}) (respectively D_{lfgu}^{b}(O_{F^{r},Xét})) in which case it is clear that Res_{q}^{q′}M^{•} is a complex in D_{lfgu}^{b}(O_{F^{r′} ,X}) (respectively D_{lfgu}^{b}(O_{F^{r′} ,Xét})).

If M^{•} is in D^{b}(O_{F^{r′} ,X}) (respectively D^{b}(O_{F^{r′} ,Xét})) then we write

Ind_{q′}^{q}M^{•} = O_{F^{r},X} ⊗_{O_{F^{r′} ,X}} M^{•} ∈ D^{b}(O_{F^{r},X}) (respectively D^{b}(O_{F^{r},Xét})).

In particular this makes sense for M^{•} a complex in D_{lfgu}^{b}(O_{F^{r′} ,X}) (respectively D_{lfgu}^{b}(O_{F^{r′} ,Xét})), in which case Ind_{q′}^{q}M^{•} is a complex in D_{lfgu}^{b}(O_{F^{r},X}) (respectively D^{b}(O_{F^{r},Xét})). To see this, consider a locally finitely generated unit O_{F^{r′} ,X}- module M. (It will suffice to consider the O_{F^{r′} ,X}-module case; the O_{F^{r′} ,Xét}-module case is proved in an identical fashion.) Then we have

O_{F^{r},X} ⊗_{O_{F^{r} ,X}} M ⟶^{∼} ⊕_{n=0}^{\frac{r′}{r} −1}O_{X}F^{rn} ⊗_{O_{X}} M.

The structural morphism (F_{X}^{r})^{∗}M → M induces maps

O_{X}F^{r} ⊗_{O_{X}} O_{X}F^{rn} ⊗_{O_{X}} M ⟶ O_{X}F^{rn+r} ⊗_{O_{X}} M, 0 ≤ n < \frac{r′}{r} − 1,

and a map

O_{X}F^{r} ⊗_{O_{X}} O_{X}F^{r′−r} ⊗_{O_{X}} M ⟶ O_{X}F^{r′}M ⟶^{∼} M.

We have to show these are isomorphisms. But this is clear, as the former maps are induced by the isomorphisms O_{X}F^{r} ⊗_{O_{X}} O_{X} ⟶^{∼} O_{X}F^{r}, and the latter map is induced by the same isomorphism followed by the structural isomorphism of M. Having shown the result for a single locally finitely generated unit module, it follows for an arbitrary complex in D_{lfgu}^{b}(O_{F^{r′} ,X}) by truncation and induction on the length of a complex.

We call the operations Res_{q}^{q′} and Ind_{q′}^{q} restriction and induction. It is clear that these operations commute with f_{+} and f^{!}, for any map f of smooth k-schemes, and that they are compatible with change of ground field.

(11.4.2) We are going to define analogous operations for constructible sheaves. To avoid confusion let us denote by D_{c,q}^{b}(X_{ét}) the bounded derived category of the category of constructible sheaves of 𝔽_{q}-vector spaces on X_{ét}.

If L^{•} is in D_{c,q′}^{b}(X_{ét}), we denote by Res_{q′}^{q}L^{•} the complex L^{•} regarded as a complex of sheaves of 𝔽_{q}-vector spaces.

If L^{•} is in D_{c,q}^{b}(X_{ét}), we set

Ind_{q}^{q′}L^{•} = L^{•} ⊗_{𝔽_{q}} 𝔽_{q′} ∈ D_{c,q′}^{b}(X_{ét}).

We call Res_{q′}^{q} and Ind_{q}^{q′} induction and restriction. It is clear that they commute with f_{!} and f^{−1} for any map f of smooth k-schemes.

Theorem 11.4.3. We have natural isomorphisms of functors

(11.4.4)  M_{ét}_{,q′} ∘Ind_{q}^{q′} ⟶^{∼} Res_{q}^{q′} ∘ M_{ét}_{,q},

(11.4.5)  Ind_{q′}^{q} ∘ M_{ét}_{,q′} ⟶^{∼} M_{ét}_{,q} ∘Res_{q′}^{q},

(11.4.6)  Sol_{ét}_{,q} ∘Ind_{q′}^{q} ⟶^{∼} Res_{q′}^{q} ∘ Sol_{ét}_{,q′},

and

(11.4.7)  Ind_{q}^{q′} ∘ Sol_{ét}_{,q} ⟶^{∼} Sol_{ét}_{,q′} ∘Res_{q}^{q′}.

Furthermore, these isomorphisms are compatible with the isomorphisms ζ : id ⟶^{∼} M_{ét} Sol_{ét} and η : id ⟶^{∼} M_{ét} Sol_{ét}.

Proof. Let L^{•} be in D_{c,q}^{b}(X_{ét}). Then we have the natural isomorphism

M_{ét}_{,q′} ∘Ind_{q}^{q′}L^{•} = \underline{RHom}_{𝔽_{q′}}^{•} (L^{•} ⊗_{𝔽_{q}} 𝔽_{q′}, O_{Xét})

⟶^{∼} \underline{RHom}_{𝔽_{q}}^{•}(L^{•}, O_{Xét}) = Res_{q}^{q′} ∘ M_{ét}_{,q} L^{•},

which is clearly compatible with inseparable base-change. This proves (11.4.4).

Suppose that L^{•} is in D_{c,q′}^{b}(X_{ét}). If I^{•} is a resolution of O_{Xét} by injective O_{F^{r},Xét}-modules (which are therefore injective as 𝔽_{q′}-sheaves) then we have a natural map

Ind_{q′}^{q} ∘ M_{ét}_{,q′} L^{•} = O_{F^{r},Xét} ⊗_{O_{F^{r′} ,Xét}} \underline{Hom}_{𝔽_{q′}}^{•} (L^{•}, I^{•})

⟶ \underline{Hom}_{𝔽_{q}}^{•}(L^{•}, I^{•}) = M_{ét}_{,q} ∘Res_{q′}^{q}L^{•}.

We have to show this map is an isomorphism. It is clear that this map is compatible with inseparable base-change; in particular we replace k by its perfect closure, and so assume that k is perfect. As usual we reduce to the case where L^{•} is concentrated in degree 0, and we set L = L^{0}.

If L is locally constant, then working étale locally we reduce to the case L = 𝔽_{q′}, when the map above gives rise to a map of left O_{F^{r},Xét}-modules,

O_{F^{r},Xét} ⊗_{O_{F^{r′} ,Xét}} O_{X} ⟶ \underline{Hom}_{𝔽_{q}}(𝔽_{q′}, O_{X})

given by sending 1 ⊗ a to the 𝔽_{q′} linear map which sends 1 ∈ 𝔽_{q′} to a, and then “forgetting” the 𝔽_{q′} linearity. This map is obtained by tensoring the natural map

𝔽_{q}[F^{r}] ⊗_{𝔽_{q}[F^{r′}]} 𝔽_{q} ⟶ Hom_{𝔽_{q}}(𝔽_{q′}, 𝔽_{q})

by O_{Xét} over 𝔽_{q}, and the latter map is an isomorphism because the automorphisms 1, F^{r}, …, F^{r′−r} of 𝔽_{q′}/𝔽_{q} are distinct, hence linearly independent.

To prove that the map we have constructed is an isomorphism for general L one makes an excision argument; combined with the fact that f_{+} and f_{!} commute with induction and restriction, this yields a reduction to the case of locally constant L, as in the proof of 11.3. This completes the proof of (11.4.5).

Now suppose that M^{•} is in D_{lfgu}^{b}(O_{F^{r′} ,X}). Then if I^{•} is as before, we have the natural isomorphism

Sol_{ét}_{,q} ∘Ind_{q′}^{q}M^{•} = \underline{Hom}_{O_{F^{r} ,Xét}}^{•} (O_{F^{r},Xét} ⊗_{O_{F^{r′} ,Xét}} M^{•}, I^{•})

⟶^{∼} \underline{Hom}_{O_{F^{r′} ,Xét}}^{•} (M^{•}, I^{•}) = Res_{q′}^{q} ∘ Sol_{ét}_{,q′} M^{•},

which again one sees is compatible with inseparable base-change. This proves (11.4.6).

To prove (11.4.7) consider M^{•} in D_{lfgu}^{b}(O_{F^{r},X}). There is a natural morphism

Ind_{q}^{q′} ∘ Sol_{ét}_{,q} = \underline{Hom}_{O_{F^{r} ,Xét}}^{•} (M^{•}, I^{•}) ⊗_{𝔽_{q}} 𝔽_{q′}

⟶ \underline{Hom}_{O_{F^{r′} ,Xét}}^{•} (M^{•}, I^{•}) = Sol_{ét}_{,q′} ∘Res_{q}^{q′}M^{•},

which is clearly compatible with inseparable base-change. Thus in order to show this is an isomorphism, we may replace k by its perfect closure, and so assume that k is perfect. As usually we may reduce to the case when M^{•} is a single module M concentrated in degree 0, and then by an excision argument together with proposition 6.7.5 to the case where M is a unit F^{r}-crystal. Finally, working étale locally and using 9.4.1, we reduce to the case M = O_{Xét}. In this case both target and source of our morphism can be identified with 𝔽_{q′} and the map between them is the identity, so we are done.

Finally, let us remark that one can see directly from the constructions involved that these natural isomorphisms are compatible with the natural isomorphisms ζ and η. □

(11.5) Now we may prove the Riemann-Hilbert correspondence between D_{c}^{b}(X_{ét}) and D_{lfgu}^{b}(O_{F^{r},X}). Indeed, this follows immediately from theorems 11.3 and 11.4.3, which provide the analogous result in the étale setting, together with the étale descent results of section 7.

Definition 11.5.1. Define the functor

M : D_{c}^{b}(X_{ét}) → D_{lfgu}^{b}(O_{F^{r},X})

to be the composition of the functors M_{ét} and Rπ_{X∗}.

Define the functor

Sol : D_{lfgu}^{b}(O_{F^{r},X}) → D_{c}^{b}(X_{ét})

to be composition of the functors π_{X}^{∗} and Sol_{ét} .

Theorem 11.5.2. The functors M and Sol are quasi-inverse to one another and hence induce an anti-equivalence of triangulated categories between D_{c}^{b}(X_{ét}) and

D_{lfgu}^{b}(O_{F^{r},X}), which respects RHom^{•} and ⊗^{𝕃} (up to a shift of d_{Y/X}), exchanges f^{−1} and f^{!}, f_{!} and f_{+} (for allowable morphisms f), and induction and restriction, and is compatible with inseparable base-change and descent.

□ (11.6) There are several problems that remain to be solved before the founda-

tions of the theory of locally finitely generated unit O_{F,X}-modules can be considered complete. Here are two, each of which is suggested by analogy with the theory of D-modules.

(11.6.1) If X is a smooth k-scheme, is the functor D^{b}(μ_{lfgu}(X)) → D_{lfgu}^{b}(O_{F,X}) an equivalence of categories? Note that in the context of holonomic D-modules the corresponding result has been proved by Beilinson [Be]: a bounded complex of Dmodules with regular holonomic cohomology sheaves is represented by a complex of regular holonomic D-modules.

(11.6.2) Theorem 11.5.2 shows that the triangulated category D_{c}^{b}(X_{ét}) admits a t-structure whose heart is equivalent to the category of locally finitely generated unit O_{F^{r},X}-modules. It is easy to check that this t-structure is not the usual one on D_{c}^{b}(X_{ét}). It would be interesting to have an intrinsic characterization of this t-structure.

In the analogous situation of D-modules, one has the notion of a perverse complex of constructible sheaves F^{•}. It is defined using the following condition (which is dual to condition (p) of [Bor, IV §22]):

(*) for any immersion i : Y → X there exists a dense open subscheme Y_{0} ⊂ Y such that the cohomology sheaves of i^{−1}F^{•}|_{Y_{0}} are locally constant and are concentrated in degrees ≤ −dimY.

Then F^{•} satisfies (*) if and only if the corresponding complex M^{•} of D-modules has its cohomology sheaves supported in non-negative degrees. (Here F^{•} and M^{•} are assumed to be related via F^{•} ⟶^{∼} \underline{RHom}_{D_{X}}^{•} (M^{•}, O_{X})[d_{X}], which is the the contravariant form of the Riemann-Hilbert correspondence.) In particular (since the Riemann-Hilbert correspondence is compatible with taking duals) F^{•} is perverse (that is, the complex M^{•} reduces to a single D-module placed in degree zero) if and only if both F^{•} and its Verdier dual DF^{•} satisfy (*).

In the context of the theory of unit O_{F^{r}}-modules developed in this paper, we can consider the same condition (*) for bounded complexes F^{•} of constructible 𝔽_{q}-sheaves on the étale site of a smooth k-scheme X, and again one finds that for any object M^{•} of D_{lfgu}^{b}(O_{F^{r},X}), the complex F^{•} = Sol(M^{•}) satisfies (*) precisely if M^{•} has its cohomology sheaves supported in non-negative degrees. However, since there is no duality in the theory of constructible 𝔽_{q}-sheaves, it is not clear how to find an analogous formulation of the dual condition that M^{•} have cohomology sheaves supported in non-positive degrees. Thus, although one can define a bounded, constructible complex of 𝔽_{q}-sheaves to be “perverse” if it is isomorphic to the image under Sol of a locally finitely generated unit O_{F^{r},X}-module, the meaning of this seems, at present, mysterious.

12. L-Functions for unit F^{r}-modules.

In this section we will define L-functions for complexes in D_{lfgu}(O_{F^{r},X}) and D_{c}(X_{ét}) when X is a smooth scheme over a finite field k. We explain the basic properties of these L-functions and prove a relative trace formula with coefficients for them.

(12.1) Let k be a finite field containing 𝔽_{q}. We will define L-functions for bounded complexes of O_{F^{r}}-modules with locally finitely generated unit cohomology sheaves on smooth k-schemes.

(12.1.1) We first define L-functions for points. Let k′ = 𝔽_{p^{s}} be a finite extension of k and write x = Spec k′. Note that 𝔽_{q} ⊂ k ⊂ 𝔽_{p^{s}}, by assumption. If M^{•} is a complex in D_{lfgu}^{b}(O_{F^{r},x}), then the restriction Res_{q}^{p^{s}}M^{•} is a complex in D_{lfgu}^{b}(O_{F^{s},x}). Consider the inverse of its structural isomorphism

ϕ_{Res_{q}^{s}M^{•}}^{−1} : M^{•} ⟶^{∼} F_{x}^{s∗}M^{•} = M^{•},

where the last equality holds because F_{x}^{s} is the identity on x. Using this automorphism of M^{•} we define

L_{u}(x, M^{•}) = ∏_{i} det_{k′}(1 − H^{i}(ϕ_{Res_{q}^{p^{s}} M^{•}}^{−1})T^{s}|H^{i}(Res_{q}^{p^{s}}M^{•}))^{(−1)^{i+1}}.

One sees checks that this lies in k[[T]], and in fact it lies in 𝔽_{q}[[T]] (as will follow from proposition 12.3 below). Also note that if 𝔽_{q} ⊂ 𝔽_{q′} ⊂ k then

L_{u}(x, Res_{q}^{q′}M^{•}) = L_{u}(x, M^{•}),

as follows directly from the definition together with transitivity of restriction.

(12.1.2) Now let X be any smooth k scheme. Denote by M(X) the set of closed points of X, and for each x ∈ M(X), write i_{x} : x → X for the natural closed immersion. For a complex M^{•} in D_{lfgu}^{b}(O_{F^{r},x}) we define the L-function L_{u}(X, M^{•}) ∈ k[[T]] by the formula

L_{u}(X, M^{•}) = ∏_{x∈M(X)} L_{u}(x, i_{x}^{!}M^{•}).

If 𝔽_{q} ⊂ 𝔽_{q′} ⊂ k then

L_{u}(X, Res_{q}^{q′}M^{•}) = L_{u}(X, M^{•}),

as follows from the corresponding property of L-functions of points. Note also that this L-function depends only on the scheme X itself (and of course the complex M^{•}), and not on the particular finite field k over which it is a scheme. (Note that usually factors in the L-function of an F -crystal are defined using the characteristic polynomial of the structural morphism, not its inverse. Our definition has been chosen so as to be compatible with the Riemann-Hilbert correspondence that we have constructed, as proposition 12.3 below will show.)

(12.2) We keep the notation of the previous paragraph. We will define Lfunctions for bounded complexes of étale sheaves of 𝔽_{q}-vector spaces with constructible cohomology sheaves on smooth k-schemes, following [De]. (In particular, the L-functions that we define will depend only on X, and not on the particular finite field k over which we view it as a scheme.)

(12.2.1) If x = Spec k′ with k′ = 𝔽_{p^{s}} a finite extension of k and L is in D_{c}^{b}(x_{ét}), we define L_{ét}(x, L^{•}) ∈ 𝔽_{q}[[T]] by the formula

L_{ét}(x, L^{•}) = ∏_{i} det_{𝔽_{q}}(1 − (F^{s})^{−1}T^{s}|H^{i}(L^{•})x̄)^{(−1)^{i+1}},

where x̄ = Spec k̄ for some algebraic closure k̄ of k containing k′, and F^{s} : a ↦ a^{p^{s}} is regarded as an automorphism of k̄ over k (with inverse (F^{s})^{−1}) which acts on the fibre H^{i}(L^{•})x̄ by functoriality. Note that if 𝔽_{q} ⊂ 𝔽_{q′} ⊂ k then

L_{ét}(X, Ind_{q}^{q′}L^{•}) = L_{ét}(X, L^{•}),

since determinants are invariant under base change.

(12.2.2) If L^{•} is in D_{c}^{b}(X_{ét}) we define the L-function L_{ét}(X, L) ∈ 𝔽_{q}[[T]] by

L_{ét}(X, L^{•}) = ∏_{x∈M(X)} L_{ét}(x, i_{x}^{−1}L^{•}).

If 𝔽_{q} ⊂ 𝔽_{q′} ⊂ k then

L_{ét}(X, Ind_{q}^{q′}L^{•}) =: L_{ét}(X, L^{•}),

as follows from the corresponding property for the L-functions of points.

Proposition 12.3. If M^{•} is in D_{lfgu}^{b}(O_{F^{r},X}) and L^{•} = Sol(M^{•}) then we have

L_{u}(X, M^{•}) = L_{ét}(X, L^{•}).

Proof. As Sol is compatible with pull-backs, we reduce immediately to the case X = x = Spec k′, with k′ a finite extension of k. Considering k′ to be the ground field rather than k does not change the L-function, and so we may also assume that k′ = k. Furthermore, since Sol interchanges induction and restriction, and L_{u} is invariant under restriction and L_{ét} is invariant under induction, we may replace 𝔽_{q} by k, and assume that k = 𝔽_{q}.

By dévissage we may assume that M^{•} = M is a single locally finitely generated unit O_{F^{r},X}-module, which is necessarily a unit F^{r}-crystal because x is a point. Thus M is simply a k vector space equipped with an automorphism ϕ_{M}.

The complex L^{•} = Sol(M) is then a single locally constant sheaf L, which when regarded as a Gal(k̄/k)-module by taking its fibre over Spec k̄ is isomorphic to the kernel of the morphism of Gal(k̄/k)-modules

\xymatrix{Hom_{k}(M, k̄) \ar[r]^{1−F^{r}∘(ϕ_{M}^{−1})^{∗}} & Hom_{k}(M, k̄)}

(where (ϕ_{M}^{−1})^{∗} denotes the morphism of Hom-groups induced by the automorphism ϕ_{M}^{−1} of M, and F^{r} is the automorphism of k̄ defined by a ↦ a^{q}). Thus we see that (F^{r})^{−1} and (ϕ_{M}^{−1})^{∗} have the same action on L.

A matrix and its adjoint have the same characteristic polynomial, and this characteristic polynomial is invariant under extension of scalars. Also, the natural morphism k̄ ⊗_{k} L → Hom_{k}(M, k̄) is an isomorphism. Putting these observations together we find that

L_{u}(Spec k, M) = det_{k}(1 − ϕ_{M}^{−1}T^{r}|M)
= det_{k}(1 − (ϕ_{M}^{−1})^{∗}T^{r}| Hom_{k}(M, k))
= det k̄(1 − (ϕ_{M}^{−1})^{∗}T^{r}| Hom_{k}(M, k̄))
= det k̄(1 − (ϕ_{M}^{−1})^{∗}T^{r}|k̄ ⊗_{k} L)
= det_{k}(1 − (ϕ_{M}^{−1})^{∗}T^{r}|L)
= det_{k}(1 − (F^{r})^{−1}T^{r}|L) = L_{ét}(Spec k, L).

□

Theorem 12.4. Let f : X → Y be a morphism of smooth k-schemes. If M^{•} is in D_{lfgu}^{b}(O_{F,X}) and L^{•} = Sol(M^{•}) then

L_{u}(X, M^{•}) = L_{ét}(X, L^{•}) = L_{ét}(Y, f_{!}L^{•}) = L_{u}(Y, f_{+}M^{•}).

Proof. The first equality follow from the previous proposition, while the middle one is a result of Deligne [De, 2.2, p. 116]. (An alternative proof is presented in the article [Cr, 5.1] of Crew. Yet another proof of the theorem (and in fact of a more general result which was conjectured by Katz) is given in [EK2] as an application of the techniques developed in this paper.) The right most equality follows from the previous proposition together with the Riemann-Hilbert correspondence, in the case that f is allowable. To see this equality in the general case, note that we may write Y as the finite disjoint union of locally closed affine sub-schemes, and that since L-functions are multiplicative with respect to distinguished triangles and are computed point by point, it suffices to verify this equality for the restriction of M^{•} to each of these affine subschemes. Since any map whose source is an affine scheme is allowable, this proves the the stated equality. □

Corollary 12.5. If X is a smooth k-scheme and M^{•} is a complex in D_{lfgu}^{b}(O_{F^{r},X}) then L_{u}(X, M^{•}) is a rational function.

Proof. If we take Y = Spec k in the previous theorem, then this follows from the equation L_{u}(X, M^{•}) = L_{u}(Y, f_{+}M^{•}), since f_{+}M^{•} is represented by a complex of k-vector spaces with bounded finite dimensional cohomology groups. □

(12.6) In [EK2] theorem 12.4 is generalized to the following result:

Theorem 12.6.1. Let k be a finite field of characteristic p containing 𝔽_{q}, and let f : Y → X be a morphism of (not necessarily smooth) k-schemes. If L^{•} is a bounded complex of W_{n}(𝔽_{q})-sheaves of finite tor-dimension and having constructible cohomology sheaves on Y_{ét} (where W_{n}(𝔽_{q}) denotes the truncated ring of Witt vectors of 𝔽_{q} of length n, for some given positive integer n) then the quotient L_{ét}(Y, L^{•})/L_{ét}(X, f_{!}L^{•}) (where the L-functions are defined in the usual manner, generalizing the definition of (12.2) in the case when n = 1), a priori an element of 1 + T W_{n}(𝔽_{q})[[T]], is in fact an element of 1 + pT W_{n}(𝔽_{q})[T].

This theorem has the following corollary:

Corollary 12.6.2. Let k be a finite field containing 𝔽_{q} and X be a smooth kscheme. Let L be a lisse W (𝔽_{q})-sheaf on X_{ét}. Then the ratio of L-functions L_{ét}(X, L)/L_{ét}(Spec k, f_{!}L), which is an element of W (𝔽_{q})[[T]], defines an invertible rigid analytic function on the closed unit p-adic disc.

Proof. For each positive integer n we see that

L_{ét}(X, L)/L_{ét}(Spec k, f_{!}L) ≡ L_{ét}(X, L/p^{n})/L_{ét}(Spec k, f_{!}L/p^{n}) (mod p)^{n},

which by the theorem lies in 1+pT W_{n}(𝔽_{q})[T]. Taking the limit as n goes to infinity proves the corollary. □

(12.6.3) This result proves part of a conjecture of Katz [Ka2, Conjecture 6.1]. In this reference, in addition to conjecturing our corollary 12.6.2, Katz conjectures that the L-function L_{ét}(X, L) (or equivalently, the quotient L_{ét}(X, L)/L_{ét}(Spec k, f_{!}L), where f : X → Spec k is the structural morphism of the k-scheme X, since L_{ét}(Spec k, f_{!}L) is manifestly a rational function) extends to a meromorphic function of T on the entire affine line. However, this result has been disproved for general p-adic lisse sheaves by Wan [Wa]. Note that from corollary 12.6.2, one sees that L_{ét}(X, L) is a meromorphic function on the closed unit disc (since as already observed, it is the product of the quotient L_{ét}(X, L)/L_{ét}(Spec k, f_{!}L), which is defined on the closed unit disc by corollary 12.6.2, and the rational function L_{ét}(Spec k, f_{!}L)).

appendix. duality and the cartier operator

(A.1) We recall some facts about duality for quasi-coherent sheaves on schemes. Let f : Y ⟶ X be a morphism of finite type between noetherian schemes which admit a dualizing complex (in the sense of [Ha, V] and [Con, I §10]); for example f could be a morphism between schemes of finite type over a field. There exits a functor f^{!} : D_{c}^{+}(X) ⟶ D_{c}^{+}(Y) with the property that if furthermore f is proper, then f^{!} is right adjoint to Rf_{∗} in the sense of derived categories: for any two complexes E^{•} in D_{qc}^{−}(Y) and F^{•} in D_{c}^{+}(X) we have a natural isomorphism

Rf_{∗}\underline{RHom}_{O_{Y}}^{•} (E^{•}, f^{!}F^{•}) ⟶^{∼} \underline{RHom}_{O_{X}}^{•} (Rf_{∗}E^{•}, F^{•}).

If we take global sections and then apply H^{0}, we obtain an isomorphism

Hom_{D(O_{Y} )}(E^{•}, f^{!}F^{•}) ⟶^{∼} Hom_{D(O_{X})}(Rf_{∗}E^{•}, F^{•}).

This adjointness yields a morphism Rf_{∗}f^{!}F^{•} → F^{•}, which we denote tr_{f}F^{•}. (In fact, in the development of the theory, the key point is to first construct this morphism in the special case when F^{•} is a dualizing complex.) This theorem is explained in [Ha, VII] and [Con, II §1], where along with many other compatibilities it is shown that if g : Z → Y is a second morphism of finite type k-schemes then there is a natural isomorphism (fg)^{!} ⟶^{∼} g^{!}f^{!}, which is compatible with traces if f and g are both proper.

This theory ought to be compatible with flat base change, in the sense that if u : X′ → X is a flat morphism of finite-type k-schemes, giving rise to the Cartesian square

\xymatrix{Y′ \ar[d]^{f′} \ar[r]^{u′} & Y \ar[d]^{f} \\ X′ \ar[r]^{u} & X,}

then there should be natural isomorphisms

u^{′∗}f^{!} ⟶^{∼} f^{′^{!}}u^{∗}.

If furthermore f (and hence f′) is proper, the diagram of natural transformations

\xymatrix{f′_{∗}f^{′!}u^{∗} \ar[r]^{tr_{f′}} \ar[d]^{∼} & u∗ \\ f′_{∗}u^{′∗}f^{!} \ar[r]^{∼} & u^{∗}f_{∗}f^{!} \ar[u]}

should commute. Such a result is proved in [Ha, VI §5] and [Con, I §10] in the case that the morphism u is residually stable (which is a very strong restriction). The case of a general flat base-change is treated in [Ver], using however different foundations. (See the remarks in the introduction of [Con, I] on the possibility of reconciling the approach of [Ver] (which is due to Deligne) with that of [Ha].)

(A.1.1) In some situations the theory described above, which is constructed via residual complexes, can be made more explicit. Suppose that f : Y → X is a regular morphism of noetherian schemes admitting a dualizing complex, in the sense that f may be factored as the composition of of a closed immersion i : Y → W which makes Y a local complete intersection in W, and a smooth morphism p : W → X. Then f^{!}O_{X} is equal to a line bundle on Y placed in degree d_{X/Y}. (Thus such a morphism f is a special instance of a Gorenstein morphism, as defined in [Ha, p. 143-4]; note that this definition is more inclusive then the definition given subsequently in [Ha, V 9.7], in which the morphism is required to be flat. The remarks of this subsection provide a special case of the program outlined by Hartshorne for such morphisms.) We denote this line bundle by ω_{Y/X} (although of course it depends on the morphism f). If f is proper than we have the trace map tr_{f} : Rf_{∗}f^{!}ω_{Y/X}[d_{X/Y}] → O_{X}.

Now is f is any complex F^{•} in D_{c}^{+}(O_{X}) there is a natural isomorphism

(A.1.2)  f^{!}F^{•} ⟶^{∼} ω_{Y/X} ⊗ 𝕃f^{∗}F^{•}[d_{Y/X}].

(Note that since f is regular, 𝕃f^{∗} has finite tor-dimension, so the right hand side makes sense as a functor on D_{c}^{+}(O_{X}).) In the case that the morphism f is the closed immersion of a local complete intersection, this isomorphism follows from [Ha, III 6.9 (a)], together with the compatibility of f^{!} with the explicit duality for finite maps provided by f^{♭}, as discussed in [Con, II §1]. When f is smooth, it follows from [Con, II 4.2.1]. In general, we may define (A.1.2) via a factorization of f; its independence of the choice of such a factorization follows from the compatibilities proved in [Con, I §8].

An important point is that if X and Y are smooth of finite type over a field k and if f is a morphism of k-schemes, then (as we see by using the usual factorization of f as the graph morphism followed by a projection) f is regular, and there is a canonical identification ω_{Y/X} = ω_{Y} ⊗ f^{∗}ω_{X}^{−1}, where ω_{X} (respectively ω_{Y}) is the canonical bundle of X (respectively Y).

Now suppose that g : Z → Y is a second regular morphism, such that the composite fg is also regular. The isomorphism (fg)^{!}O_{X} ⟶^{∼} g^{!}f^{!}O_{X} ⟶^{∼} g^{!}ω_{Y/X} ⟶^{∼} ω_{Z/Y} ⊗_{O_{Z}} g^{∗}ω_{Y/X} provides the sense in which formation of ω_{Y/X} is compatible with composition. Furthermore, we then have a commutative diagram

\xymatrix{(fg)^{!}F^{•} \ar[r]^{∼} \ar[dd]^{∼} & ω_{Z/X} ⊗_{O_{Z}} 𝕃(fg)^{∗}F^{•}[d_{Z/X}] \ar[d]^{∼} \\  & ω_{Z/Y} ⊗_{O_{Z}} g^{∗}ω_{Y/X} ⊗_{O_{Z}} 𝕃g^{∗}𝕃f^{∗}F^{•}[d_{Z/Y}][d_{Y/X}] \ar[d]^{∼} \\ g^{!}f^{!}F^{•} \ar[r]|{∼} & ω_{Z/Y} ⊗_{O_{Z}} 𝕃g^{∗}(ω_{Y/X} ⊗_{O_{Y}} 𝕃f^{∗}F^{•}[d_{Y/X}])[d_{Z/Y}].}

One may use the right-hand side of (A.1.2) to define f^{!}F^{•} for any object F^{•} of D_{qc}^{+}(O_{X}), and then in the case when f is proper, use the projection formula to define tr_{f}F^{•} (as in [Ha, VII §4] and [Con, II §4]). A limit argument as in the proof of [Con, II §4.1.1] then shows that the resulting morphism

Rf_{∗}\underline{RHom}_{O_{Y}}^{•} (E^{•}, f^{!}F^{•}) → \underline{RHom}_{O_{X}}^{•} (Rf_{∗}E^{•}, F^{•})

is a quasi-isomorphism for any two complexes E^{•} in D_{qc}^{−}(O_{Y}) and D_{qc}^{+}(O_{X}). Furthermore this definition of tr_{f} is compatible with that defined via residual complexes for complexes in D_{c}^{b}(O_{X}), when one compares the two via isomorphism (A.1.2). (This compatibility in the smooth case is proved in [Con, II 4.3.1]. In the case when f is the closed immersion of a local complete intersection, it follows from [Ha, III 6.9 (c)]. Note that although both trace maps are defined for complexes in D_{qc}^{+}(O_{X}), the necessary compatibility in the smooth case is only proved in [Con, II] for bounded complexes.)

The isomorphism (A.1.2), and the resulting trace map in the case that f is proper, is also compatible with flat base-change. To see this, one treats the case when f is smooth and the case when f is the closed immersion of a local complete intersection separately. For the first case the base-change result follows from [Con, II 2.4.1]. In the case of the closed immersion of a local complete intersection, it follows from [Ha, III 6.3, 6.6 (2)].

It would be useful to fully realize the program outlined in [Ha, III p.143-144] for Gorenstein morphisms. (These would include as a special case morphisms which are regular (in the above sense) locally on Y, which is a more flexible condition than the global factorization that we have required.) The case of flat Gorenstein morphisms is covered by [Con, II].

(A.1.3) If f : Y → X as above is in fact a finite morphism then the compatibility of duality, as defined via residual complexes, with the explicit duality for finite morphisms of [Ha, III §6], yields a natural isomorphism

f^{!}F^{•} = f^{♭}F^{•} = \underline{RHom}_{O_{X}}^{•} (f_{∗}O_{Y} , F^{•}).

Thus if f is the closed immersion of a local complete intersection then (A.1.2) specializes to an isomorphism

(A.1.4)  \underline{RHom}_{O_{X}}^{•} (f_{∗}O_{Y} , F^{•}) ⟶^{∼} ω_{Y/X} ⊗ 𝕃f^{∗}F^{•}[d_{Y/X}].

In this situation the line bundle ω_{Y/X} can be identified with the top exterior product of the conormal bundle of Y in X, and the isomorphism (A.1.4) is then explicitly constructed using Koszul complexes. More precisely, one uses Koszul complexes to construct an isomorphism

\underline{Ext}_{O_{X}}^{d_{X/Y}} (f_{∗}O_{Y} , F) ⟶^{∼} ω_{Y/X} ⊗_{O_{Y}} f^{∗}F

for any sheaf of O_{X}-modules F (the fundamental local isomorphism of [Ha, III §7] and [Con, I §6]), and then the isomorphism (A.1.4) is deduced by the homological algebra of [Ha, I 7.4] and [Con, I 2.1]. This explicit description of (A.1.4) is compatible with flat base-change; to see this, it is enough (by the construction of [Ha, I 7.4] and [Con, I 2.1]) to observe that the property of being acyclic for the derived functors appearing on each side of (A.1.4) is preserved by flat base-change. But since locally both derived functors may be computed via Koszul complexes, this is clear.

Let us close this subsection by making a remark on notation. If I is an ideal cutting out a closed subscheme Y of X, and f : Y → X denotes the closed immersion, we will sometimes write F[I] = \underline{Hom}_{O_{X}} (f_{∗}O_{Y} , F) = H^{0}(f^{♭}F) for the sheaf of sections of F annihilated by I. We regard F[I] as a sheaf of O_{Y} -modules. Given this convention, we should write f_{∗}F[I] is we wish to regard F[I] as a sub-sheaf of the sheaf F of O_{X}-modules; however, no confusion should be caused if we sometimes abuse notation and denote this sheaf simply by F[I] also.

(A.2) For the remainder of the appendix we assume that k is a field of characteristic p. Suppose that f : Y → X is a morphism of smooth k-schemes. For any integer r we may form the r^{th} relative Frobenius diagram of Y over X :

\xymatrix{Y \ar[r]^{F_{Y /X}^{(r)}} \ar[dr]^{f} & Y^{(r)} \ar[r]^{F_{X}^{r′}} \ar[d]^{f^{(r)}} & Y \ar[d]^{f} \\  & X \ar[r]^{F_{X}^{r}} & X.}

This is a commutative diagram defined so that the right square is Cartesian, and the composite of the two upper horizontal arrows is equal to F_{Y}^{r} . (Note that the fibre product which we denote by Y^{(r)} is frequently denoted by Y^{(p^{r})}. Since we always deal with a fixed prime p, our notation should not cause any confusion.

In particular if we replace Y by X (for any smooth k-scheme X) and X by Spec k, then we may write F_{X}^{r} as the composition of the r^{th} power of the relative Frobenius F_{X/k}^{r} : X → X^{(r)} (which is a morphism of k-schemes) and the base-change F_{k}^{r}′ of the r^{th} power of the Frobenius morphism F_{k}^{r} : Spec k → Spec k. The morphism F_{X/k}^{r} is a finite flat map (since X is smooth over k), but the morphism F_{k}^{r} need not be so (it is finite flat if and only if k has finite degree over k^{p}). However it is an affine morphism, and k is free over k^{p}, which implies that F_{k}^{r}′ is an affine map such that F_{X/k∗}^{r}O_{X′} is free over O_{X}. Thus F_{X}^{r} = F_{X/k}^{r}F_{k}^{r}′ is affine, with the property that F_{X∗}^{r}O_{X′} is locally free over O_{X}.

Now return to the situation of a morphism f : Y → X between smooth kschemes. Since X and Y are of finite type over k, we may find a subfield k_{1} of k which is finitely generated over 𝔽_{p}, smooth schemes X_{1} and Y_{1} over k_{1}, and a morphism f_{1} : Y_{1} → X_{1} such that the schemes X and Y and the morphism f : Y → X are obtained by base-changing the schemes X_{1} and Y_{1} and the morphism f_{1} : X_{1} → Y_{1} from the field k_{1} to the field k.

Since the morphism F_{X_{1}}^{r} and F_{Y_{1}}^{r} are finite flat (k_{1} being finite over k_{1}^{p}) we see that the morphism F_{Y_{1}/X_{1}}^{(r)} is finite (although it is not flat in general). Since the

morphism F_{Y/X}^{(r)} is obtained by base-changing from k_{1} to k, it is also finite. Now consider the isomorphisms

f^{!}O_{X} ⟶^{∼} F_{Y/X}^{(r)!}f^{(r)!}O_{X} ⟶^{∼} F_{Y/X}^{(r)!}f^{(r)!}F_{X}^{r∗}O_{X} ⟶^{∼} F_{Y/X}^{(r)!}F_{X}^{r}^{′∗}f^{!}O_{X}

Since F_{Y/X}^{(r)} is finite, we may apply adjointness of F_{Y/X∗}^{(r)} and F_{Y/X}^{(r)!} and obtain a map

F_{Y/X∗}^{(r)}f^{!}O_{X} ⟶ F_{X}^{r}′^{∗}f^{!}O_{X}.

Since X, Y are smooth, we may rewrite this (after shifting) as

C_{Y/X}^{(r)} : F_{Y/X∗}^{(r)}ω_{Y/X} ⟶ F_{X}^{r′∗}ω_{Y/X}.

We refer to this morphism as the r^{th} relative Cartier operator of Y over X.

(A.2.1) Suppose that f : Y → X is a smooth morphism of k-schemes. Then the r^{th} relative Cartier operator coincides with the restriction to top forms of the r^{th} iterate of the usual Cartier operator (as described for example in [DI]). For in this case F_{Y/X}^{(r)} is a finite morphism of smooth X-schemes, and so C_{Y/X}^{(r)} : F_{Y/X}^{(r)}ω_{Y/X} → F_{X}^{r′∗}ω_{Y/X} is given by the usual trace map on top forms.

(A.2.2) Suppose that f : Y → X is a closed immersion of smooth k-schemes. Then the relative Cartier operator is calculated in the last paragraph of page 96 of [Lyu] (although not under this name). In this subsection we make several useful computations involving the r^{th} relative Frobenius diagram of the closed immersion f, which in particular make apparent the connection with the calculations of [Lyu].

Let M be an O_{X}-module. The isomorphism f^{!}M ⟶^{∼} F_{Y/X}^{(r)!}f^{(r)!}M yields a morphism

\underline{Ext}^{d_{X/Y}} (f_{∗}O_{Y} , M) → \underline{Ext}^{d_{X/Y ^{(r)}}} (f_{∗}^{(r)}O_{Y^{(r)}}, M)[I];

this is an edge map of the spectral sequence

\xymatrix{\underline{Ext}_{O_{Y ^{(r)}}}^{p} (F_{Y/X∗}^{(r)}O_{Y} , \underline{Ext}_{O_{X}}^{q} (f_{∗}^{(r)}O_{Y^{(r)}}, M)) \ar@{=>}[r] & \underline{Ext}_{O_{X}}^{p+q}(f_{∗}O_{Y} , M).}

We will describe this edge map in terms of the fundamental local isomorphism.

Let a_{1}, …, a_{s} be a regular sequence of sections which (locally) define Y. Then a_{1}^{q}, …, a_{s}^{q} define Y^{(r)}. We may compute \underline{Ext}_{O_{X}}^{d_{X/Y}} (f_{∗}O_{Y} , M) as the top cohomology of the complex K^{•}(a, M). (We are following the notational conventions of [Con, I §1]. This is the complex whose n^{th} term is equal to \underline{Hom}(Λ^{n}O_{X}^{s}, M), and whose n^{th} differential is given by the formula (letting e_{1}, … e_{s} denoting the standard basis of O_{X}^{s})

d^{n}α(e_{i_{1}} ∧ ⋯ e_{i_{p+1}}) = ∑_{j=1}^{p+1}(−1)^{p+j}a_{i_{j}}α(e_{i_{1}} ∧ ⋯ ∧ ê_{i_{j}} ∧ ⋯ ∧ e_{i_{p+1}}).

Similarly, we may compute \underline{Ext}_{O_{X}}^{d_{X/Y ^{(r)}}} (f^{(r)}O_{Y^{(r)}}, M) as the top cohomology of K^{•}(a^{p}, M). Given this, it is not hard to see that the above edge homomorphism is given by the taking the top cohomology of the map of complexes

ι : K^{•}(a, M) → K^{•}(a^{p}, M)

defined in degree n by the formula

ι(α)(e_{i_{1}} ∧ ⋯ ∧ e_{i_{n}}) = a_{1}^{q−1} ⋯ a_{s}^{q−1}α(e_{i_{1}} ∧ ⋯ ∧ e_{i_{n}}).

Thus the following diagram (in which the horizontal arrows are given by the fundamental local isomorphism and the left hand vertical arrow is the above edge map) commutes:

\xymatrix{\underline{Ext}_{O_{X}}^{d_{Y /X}} (f_{∗}O_{Y} , M) \ar[r]^{∼} \ar[d] & ω_{Y/X} ⊗_{O_{Y}} f^{∗}M \ar[d]^{{(da_{1}∧···∧das)^{−1}⊗m \atop ↦a{q−1 \atop 1}···a{q−1 \atop s}(da{p \atop 1}∧···∧da^{p}_{s})^{−1}⊗m}} \\ \underline{Ext}_{O_{X}}^{d_{X/Y ^{(r)}}} (f_{∗}^{(r)}O_{Y^{(r)}}, M)[I] \ar[r] & (ω_{Y^{(r)}/X} ⊗_{O_{Y ^{(r)}}} f^{(r)∗}M)[I] \ar[d]^{∼} \\  & (F_{X}^{r′∗}ω_{Y/X} ⊗_{Y^{(r)}} f^{(r)∗}M)[I].}

In the particular case that M = O_{X}, we deduce that the composite of the right hand vertical arrows is the map

ω_{Y/X} → F_{Y/X}^{(r)!}F_{X}^{r′∗}ω_{Y/X}

which induces

C_{Y/X}^{(r)} : F_{Y/X∗}^{(r)}ω_{Y/X} → F_{X}^{r′∗}ω_{Y/X}

by adjointness. Thus C_{Y/X}^{(r)} is given by the formula

(da_{1} ∧ ⋯ ∧ da_{s})^{−1} ↦ a_{1}^{q−1} ⋯ a_{s}^{q−1} ⊗ (da_{1} ∧ ⋯ ∧ da_{s})^{−1}.

There is one more construction in the context of the closed immersion f that we will explain. Suppose that N is an O_{Y} -module. Then we may tensor C_{Y/X}^{(r)} with F_{X}^{r′∗}N to obtain a morphism

F_{Y/X∗}^{(r)}(ω_{Y/X} ⊗_{O_{Y}} F_{Y}^{r∗}N) ⟶^{∼} F_{Y/X∗}^{(r)}ω_{Y/X} ⊗_{O_{Y ^{(r)}}} F_{X}^{r′∗}N

⟶ F_{X}^{r′∗}ω_{Y/X} ⊗_{O_{Y ^{(r)}}} F_{X}^{r′∗}N ⟶^{∼} F_{X}^{r′∗}(ω_{Y/X} ⊗_{O_{Y}} N).

Adjointness yields a morphism

ω_{Y/X} ⊗_{O_{Y}} F_{Y}^{r∗}N → (F_{X}^{r′∗}(ω_{Y/X} ⊗_{O_{Y}} N))[I].

The above formula for C_{Y/X}^{(r)} yields the explicit formula

(da_{1} ∧ ⋯ ∧ da_{s})^{−1} ⊗ m ↦ a_{1}^{q−1} ⋯ a_{s}^{q−1} ⊗ (da_{1} ∧ ⋯ ∧ da_{s})^{−1} ⊗ m

for this morphism. This map is an isomorphism (as follows either from the isomorphism [Ha, III 6.9 (a)], with F^{•} = ω_{Y^{(r)}/X} = F_{X}^{r′∗}ω_{Y/X} and G^{•} = F_{X}^{r′∗}N, or from the calculations at the bottom of page 96 and the top of page 97 of [Lyu]).

(A.2.3) The following list of properties uniquely determines the relative Cartier operator:

(i) Etale morphisms: If f : Y → X is étale, so that ω_{Y/X} = O_{Y} and F_{Y/X}^{(r)}

is an isomorphism of schemes, C_{Y/X}^{(r)} : F_{Y/X∗}^{(r)}O_{Y} → O_{Y^{(r)}} is the inverse of the

isomorphism O_{Y^{(r)}} → F_{Y/X∗}^{(r)}O_{Y} defined by the isomorphism F_{Y/X}^{(r)}. (This follows from (A.2.1).)

(ii) Products: If Z is a smooth k-scheme, we may form the product morphism

f × id_{Z} : Y × Z → X × Z

(all the products being taken over 𝔽_{q}). Then there is a natural identification of (Y × Z)^{(r)} with Y^{(r)} × Z.

Letting p_{1} (respectively p′_{1}) denote the projection of the product Y × Z (respectively (Y ×Z)^{(r)} ⟶^{∼} Y^{(r)}×Z) onto its first factor, there is a canonical identification ω_{Y ×Z/X×Z} = p_{1}^{∗}ω_{Y/X}. Then C_{Y ×Z/X×Z}^{(r)} is given by

F_{Y ×Z/X×Z∗}^{r}ω_{Y ×Z/X×Z} ⟶^{∼} (F_{Y/X}^{(r)} × id_{Z})_{∗}p_{1}^{∗}ω_{Y/X} ⟶^{∼} p′_{1}^{∗}F^{r}_{Y/X_{∗}}ω_{Y/X}

⟶^{p_{1}^{′∗}C_{Y /X}^{(r)}} p′_{1}^{∗}F_{X}^{r}′^{∗}ω_{Y/X} ⟶^{∼} (F_{X}^{r}′ × F_{Z}^{r})^{∗}p_{1}^{∗}ω_{Y/X} ⟶^{∼} F^{r}′_{X×Z}^{∗}ω_{Y ×Z/X×Z}.

(This follows from the compatibility of duality with flat base-change.)

(iii) Composition: Suppose that f : Y → X and g : Z → Y are two morphisms of smooth k-schemes. Then we may amalgamate the relative Frobenius diagrams of Y over X, of Z over X and of Z over Y into the following diagram:

\xymatrix{Z \ar[r]^{F_{Z/Y}^{(r)}} \ar[dr]^{g} & Z′′ \ar[r]^{F_{Y /X}^{(r)′}} \ar[d]^{g_{Y}^{(r)}} & Z′ \ar[r]^{F_{X}^{r′′}} \ar[d]^{g^{(r)}} & Z \ar[d]^{g} \\  & Y \ar[r]^{F_{Y /X}^{(r)}} \ar[dr]^{f} & Y^{(r)} \ar[r]^{F_{X}^{r′}} \ar[d]^{f^{(r)}} & Y \ar[d]^{f} \\  &  & X \ar[r]^{F_{X}^{r}} & X.}

The relative Cartier operator

C_{Z/X}^{(r)} : F_{Z/X∗}^{(r)}ω_{Z/X} → F_{X}^{r}′′^{∗}ω_{Z/X}

is determined by the relative Cartier operators C_{Y/X}^{(r)}, C_{Z/Y}^{(r)} , as follows

F_{Z/X∗}^{(r)}ω_{Z/X} ⟶^{∼} F_{Y/X∗}^{(r)′}F_{Z/Y ∗}^{(r)}(ω_{Z/Y} ⊗ g^{∗}ω_{Y/X})

⟶^{∼} F_{Y/X∗}^{(r)′}F_{Z/Y ∗}^{(r)}(ω_{Z/Y} ⊗ F_{Z/Y}^{(r)∗}g_{Y}^{(r)∗}ω_{Y/X})

⟶^{∼} F_{Y/X∗}^{(r)′}(F_{Z/Y ∗}^{(r)}ω_{Z/Y} ⊗ g_{Y}^{(r)}^{∗}ω_{Y/X})

⟶^{F_{Y /X∗}^{(r)′}(C_{Z/Y}^{(r)} ⊗1)} F_{Y/X∗}^{(r)′}(F_{Y}^{r} ^{′∗}ω_{Z/Y} ⊗ g_{Y}^{(r)}^{∗}ω_{Y/X})

⟶^{∼} F_{Y/X∗}^{(r)′}(F_{Y/X}^{(r)′∗}F_{X}^{r}^{′′∗}ω_{Z/Y} ⊗ g_{Y}^{(r)}^{∗}ω_{Y/X})

⟶^{∼} F_{X}^{r}^{′′∗}ω_{Z/Y} ⊗ F_{Y/X∗}^{(r)′}g_{Y}^{(r)}^{∗}ω_{Y/X}

⟶^{∼} F_{X}^{r′′}^{∗}ω_{Z/Y} ⊗ g^{(r)}^{∗}F_{Y/X∗}^{(r)}ω_{Y/X}

⟶^{1⊗g^{(r)}^{∗}C_{Y /X}^{(r)}} F_{X}^{r′′}^{∗}ω_{Z/Y} ⊗ g^{(r)}^{∗}F_{X}^{r′}^{∗}ω_{Y/X}

⟶^{∼} F_{X}^{r}^{′′∗}ω_{Z/Y} ⊗ F_{X}^{r}^{′′∗}g^{∗}ω_{Y/X}

⟶^{∼} F_{X}^{r′′∗}(ω_{Z/Y} ⊗ g^{∗}ω_{Y/X}).

(This follows from the compatibility of duality with composition.)

(iv) Embedding a divisor: If f : Y → X is the closed immersion of a divisor, let O_{X}(Y) denote the invertible sheaf corresponding to Y (the inverse of the ideal sheaf of Y). Then Y^{(r)} is the divisor corresponding to the invertible sheaf O_{X}(Y)^{⊗q}. There is a natural isomorphism ω_{Y/X} = f^{∗}O_{X}(Y) = O_{X}(Y)/O_{X}, and

F_{X}^{r}′^{∗}ω_{Y/X} ⟶^{∼} F_{X}^{r}′^{∗}f^{∗}O_{X}(Y) ⟶^{∼} f^{(r)∗}F_{X}^{r}^{∗}O_{X}(Y)

⟶^{∼} f^{(r)∗}O_{X}(Y^{(r)}) ⟶^{∼} O_{X}(Y^{(r)})/O_{X}.

The map F_{Y/X}^{(n)} is the closed immersion of Y into Y^{(r)}. In this situation, the relative Cartier isomorphism

C_{Y/X}^{(r)} : F_{Y/X∗}^{(r)}(O_{X}(Y)/O_{X}) → O_{X}(Y^{(r)})/O_{X}

is the map induced by the inclusion

O_{X}(Y) → O_{X}(Y)^{⊗q} = O_{X}(Y^{(r)}).

(This follows from (A.2.2), if one works in terms of a local equation for Y.)

(v) Mapping 𝔾_{m} to a point: In the particular case of Y = 𝔾_{m} = Spec k [t, t^{−1}], the differential dt/t forms a global basis for the invertible sheaf ω_{𝔾_{m}}, allowing us to identify this sheaf with O_{𝔾_{m}}, and under this identification the Cartier operator becomes the trace of the Frobenius F_{Y/k}^{r} : O_{𝔾_{m}} → O_{𝔾_{m}}. (This follows from (A.2.1).)

Note that (i) – (v) determine the Cartier operator for any morphism Y ⟶ X of smooth k schemes. Indeed any such map is the composite of a closed immersion and a smooth map (use the usual factorization via the graph), and any smooth map factors étale locally into a composite of projections 𝔾_{m} × X ⟶ X.

References

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[Bor] Borel, A., et al., Algebraic D-modules, Perspectives in Mathematics 2, Academic Press, 1987.

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