A Short Course on Galois Cohomology

William Stein Spring 2010

Contents

1 Preface  3
2 Introduction  4
3 G-modules  5
4 H^{q} and Ext  7
5 Morphisms of pairs  10
5.1 Morphism of pairs . . . . . . . . . . . . . . . . . . . . . . . . 10
5.2 Shapiro’s lemma . . . . . . . . . . . . . . . . . . . . . . . . . 11
6 Inflation and restriction  12
7 Inner automorphisms  12
8 An application of inner automorphisms  13
9 The restriction-inflation sequence  14
10 Cohomology sets  15
11 Long exact sequence of cohomology sets  17
12 Homology  18
13 Tate cohomology groups  19
14 Complete resolution of G  22

15 Corestriction  23
16 Cup Product  25
16.1 The Definition . . . . . . . . . . . . . . . . . . . . . . . . . . 25
16.2 Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
16.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
16.4 Cohomology of a Cyclic Group . . . . . . . . . . . . . . . . . 28
17 Galois Cohomology  29
17.1 The Definition . . . . . . . . . . . . . . . . . . . . . . . . . . 29
17.2 Infinite Galois extensions . . . . . . . . . . . . . . . . . . . . 30
17.3 Some Galois Modules . . . . . . . . . . . . . . . . . . . . . . . 31
17.4 The Additive and Multiplicative Groups of a Field . . . . . . 32
18 Kummer Theory  33
18.1 Kummer Theory of Fields . . . . . . . . . . . . . . . . . . . . 33
18.2 Kummer Theory for an Elliptic Curve . . . . . . . . . . . . . 35
19 Brauer Groups  37
19.1 The Definition . . . . . . . . . . . . . . . . . . . . . . . . . . 37
19.2 Some Motivating Examples . . . . . . . . . . . . . . . . . . . 38
19.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
19.4 Brauer Groups and Central Simple Algebras . . . . . . . . . . 40
20 Galois Cohomology of Abelian Varieties  42
20.1 Principal Homogenous Spaces for Abelian Varieties . . . . . . 42
20.2 Galois Cohomology of Abelian Varieties over Finite Fields . . 43
21 Duality  46
21.1 Duality over a Local Field . . . . . . . . . . . . . . . . . . . . 46
21.1.1 Example: n-torsion on an elliptic curve . . . . . . . . 47
21.2 Duality over a Finite Field . . . . . . . . . . . . . . . . . . . . 48
22 A Little Background Motivation for Étale Cohomology  49
22.1 Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
22.2 Étale Cohomology . . . . . . . . . . . . . . . . . . . . . . . . 50
23 Étale Cohomology over a DVR  51
23.1 Discrete Valuation Rings . . . . . . . . . . . . . . . . . . . . . 51
23.2 Galois Groups associated to DVR’s . . . . . . . . . . . . . . . 51
23.3 Galois Modules over a DVR . . . . . . . . . . . . . . . . . . . 52

23.4 The Natural Functors . . . . . . . . . . . . . . . . . . . . . . 53
23.5 Cohomology of Galois Modules over a DVR . . . . . . . . . . 54
24 The Étale Topology  54
24.1 Special Cases of Interest . . . . . . . . . . . . . . . . . . . . . 56
24.2 Étale Coverings . . . . . . . . . . . . . . . . . . . . . . . . . . 56
24.3 Étale Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
24.4 Direct and Inverse Image Functors . . . . . . . . . . . . . . . 58
24.5 Stalks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
24.6 Pullback and Pushforward of Étale Sheaves . . . . . . . . . . 60
25 Étale Cohomology  61
26 Galois Modules and Étale Sheaves  62
26.1 The Spectrum of a Field . . . . . . . . . . . . . . . . . . . . . 62
26.2 The Spectrum of a DVR . . . . . . . . . . . . . . . . . . . . . 63
27 Étale Cohomology over a DVR  65
28 The Multiplicative Group over a DVR  71
29 Étale Cohomology of Abelian Varieties  73

1 Preface

These are the notes from a one-quarter course on Galois cohomology, which the author taught at University of Washington in 2010. They are mostly based on the best parts of [AW67] and [Ser67].

Acknowledgement: Sebastian Pancratz (of Cambridge University) typed up the first 9 lectures. The course was attended by Robert Bradshaw, Amelia Chen, Alison Deines, Ralph Greenberg, Jacob Lewis, Robert Miller, Amy Supple, and Wenhan Wang who all made numerous comments. The author learned Galois cohomology mainly in person from Hendrik Lenstra, to whom he’s greatly indebted.

2 Introduction

Number Theory is the study of G = Gal(ℚ̄/ℚ), the group of automorphisms of the algebraic closure ℚ̄, and the sets G naturally acts on. The following question, for example, is an open problem: is every finite group a quotient of G?

Galois cohomology involves studying the group G by applying homological algebra. This provides a natural way to classify objects, e.g. twists of a curve, and linearizes problems by defining new invariants, revealing previously hidden structure.

This course will consists of mainly two parts, one on group cohomology in greater generality and one on Galois cohomology. In the first part, we will apply homological algebra to groups, solving problems like the following: given a group G acting on an abelian group A, find all extensions of G by A, that is, exact sequences

0 → A → E → G → 1,

where we write both 0 and 1 for the trivial group with one element. In the second part, we will apply group cohomology to number theory. This is a very important tool, vital to most advances in algebraic number theory in the last thirty years.

We now give further details of the course structure. The first part will cover the basic theory, involving essentially no number theory:

• G-modules

• cocycles, coboundaries

• basic homological algebra

• dimension shifting

• inflation/ restriction

• cup products, etc.

We will include most proofs in this part. In the second part, on Galois cohomology, we will specialize to number theory, focussing more on examples and including fewer proofs. The topics we will consider are:

• Profinite groups and their cohomology; topological groups

• Hilbert 90, and applications • Kummer theory

• Descent of the field of definitions of a variety

• Twists of algebraic curves

• Brauer groups, which are are classes of simple algebras

• Elliptic curves

• Duality: Tate local duality, and Pouitou–Tate global duality

• Lang’s theorem

• Shafarevich–Tate group

• Étale cohomology

3 G-modules

We will follow Chapter VII of Serre’s Local Fields for a while. Definition 1. Suppose G is any group and A is an abelian group with a G-action, that is, a map G × A → A such that

1.a = a, s.(a + a′) = s.a + s.a′, (st).a = s.(t.a)

Let Λ = ℤ[G], the abelian group of formal finite sums of elements of G. This makes A a Λ−module, which we will also refer to as a G-module. Remark 2. The category Mod_{G} of G-modules is an abelian category. Thus, we have direct sums, kernels, co-kernels, etc. Definition 3. A sequence A ⟶^{f} B ⟶^{g} C, where A, B, and C are objects in Mod_{G} and f, g are morphisms, is exact at B if Im(f) = ker(g). A sequence 0 → A → B → C → 0 is a short exact sequence if it is exact at A, B, and C. Definition 4. The map Mod_{G} → 𝔄𝔟 from Mod_{G} to the category 𝔄𝔟 of abelian group given by A ↦ A^{G}, which is the G-invariant subgroup of A,

A^{G} = {a ∈ A : ∀g ∈ G g.a = a}

is a functor.

Let H^{0}(G, A) = A^{G}.

Proposition 5. If 0 → A ⟶^{f} B ⟶^{g} C → 0 is an exact sequence of G-modules, then the sequence

0 → A^{G} → B^{G} → C^{G}

is exact at A^{G} and B^{G}. We say the functor A ↦ A^{G} is left exact.

Proof. Suppose that b ∈ B^{G} and g(b) = 0. Then there exists a ∈ A such that f(a) = 0. But for any s ∈ G,

f(s.a) = s.f(a) = s.b = b

so f(s.a) = f(a), hence a ∈ A^{G}, since f is injective. The proof can be completed in a similar way.

Definition 6. Formally, one can define functors H^{q}(G, −) : Mod_{G} → 𝔄𝔟 for all q ≥ 0 such that if 0 → A → B → C → 0 is exact then we have a long exact sequence

0 → A^{G} → B^{G} → C^{G} → H^{1}(G, A) → H^{1}(G, B) → H^{1}(G, C)

→ H^{2}(G, A) → H^{2}(G, B) → H^{2}(G, C) → …

Definition 7. A (co-variant) functor F : Mod_{G} → 𝔄𝔟 is exact if for every short exact sequence 0 → A → B → C → 0 the sequence 0 → F(A) → F(B) → F(C) → 0 is also exact. If F is contravariant, we similarly require that the sequence 0 ← F(A) ← F(B) ← F(C) ← 0 is exact.

Definition 8. A G-module A is projective if the functor Hom_{G}(A, −) is exact. A is called injective if the (contravariant) functor Hom_{G}(−, A) is exact.

Definition 9. A is induced if A ≅ Λ ⊗_{ℤ} X for some X ∈ 𝔄𝔟, that is, if there is a subgroup X ⊂ A such that A = ⨁_{s∈G} s.X. A is co-induced if A ≅ Hom_{ℤ}(Λ, X) for some X ∈ 𝔄𝔟.

Remark 10. If G is a finite group, then Hom_{ℤ}(Λ, X) ≅ Λ⊗_{ℤ} via the isomorphism f ↦ ∑_{s∈G} s ⊗ f(s) for all X ∈ 𝔄𝔟, so that the notions of induced and co-induced G-modules coincide.

Lemma 11. Every A ∈ Mod_{G} is a quotient of an induced module.

Proof. Let A_{0} denote the group underlying A. Then t.(s ⊗ a) = (ts) ⊗ a. Thus,

Λ ⊗ A_{0} ↠ A

Definition 12. The functors H^{q}(G, −), for q ≥ 0, are called right derived functors of the left exact functor A ↦ A^{G}. The construction is functorial in exact sequences, that is, from a short exact sequence 0 → A → B → C → 0 we obtain a long exact sequence

0 → H^{0}(G, A) → H^{0}(G, B) → H^{1}(G, C)

⟶^{δ} H^{1}(G, A) → H^{1}(G, B) → H^{1}(G, C) ⟶^{δ} …

with connecting homomorphisms δ. H^{q}(G, −) is a cohomological δ-functor.

Theorem 13. H^{q}(G, −) is the unique δ-functor such that

(i) H^{0}(G, A) = A^{G};

(ii) if A is injective then, for all q ≥ 1, H^{q}(G, A) = 0.

Thus, everything we can prove about the functor H^{q}(G, −) must follow from the two properties above.

4 H^{q} and Ext

Noting that A^{G} = Hom_{G}(ℤ, A), we deduce that H^{q}(G, A) = Ext_{G}^{q}(ℤ, A). Thus, we can in principle compute the group H^{q}(G, A) as follows:

(i) Find a resolution of ℤ by projective G-modules P_{0}, P_{1}, …

… → P_{1} → P_{0} → ℤ → 0.

(ii) Apply the functor Hom_{G}(−, A) to obtain groups K^{i} = Hom_{G}(P_{i}, A) and the complex

0 → Hom_{G}(P_{0}, A) ⟶^{d_{0}} Hom_{G}(P_{1}, A) ⟶^{d_{1}} Hom_{G}(P_{2}, A) ⟶^{d_{2}} ….

Then H^{q}(G, A) = ker(d_{q})/ Im(d_{q−1}).

We will later carry out the above procedure to obtain a very explicit representation.

Suppose that G acts diagonally on P_{i}, that is, that P_{i} = ℤ[G^{i+1}] = ℤ[G × ⋯ × G]. Then the map d: P_{i+1} → P_{i} is given by

d(g_{0}, …, g_{i}) = ∑_{j=0}^{i}(−1)^{j}(g_{0}, …, g_{j−1}, g_{j+1}, …, g_{i}).

Definition 14. A G-module A ∈ Mod_{G} is relatively injective if it is a direct factor of a co-induced module, i.e., if A⊕B is co-induced for some B ∈ Mod_{G}.

Proposition 15. If A ∈ Mod_{G} is relatively injective then H^{q}(G, A) = 0 for all q ≥ 1.

Proof. It suffices to consider co-induced G-modules, since we can then deduce the general case from the identity H^{q}(G, A⊕B) = H^{q}(G, A)⊕H^{q}(G, B). So assume that A is co-induced, that is, A ≅ Hom_{ℤ}(Λ, X) for some X ∈ 𝔄𝔟. For B ∈ Mod_{G},

Hom_{ℤ}(B, X) ≅ Hom_{G}(B, A), f ↦ (b ↦ (s ↦ f(s.b))).

From the above it follows that Hom_{G}(P_{i}, A) = Hom_{ℤ}(P_{i}, X), so

H^{q}(G, A) = Ext_{ℤ}^{q}(ℤ, X) = 0

for q ≥ 1. Moreover, Hom_{ℤ}(ℤ, X) ≅ X so the functor Hom_{ℤ}(ℤ, −) is exact.

Corollary 16 (Dimension shifting). If 0 → A → A^{∗} → A′ → 0 is exact with A^{∗} co-induced then

H^{q}(G, A′) = H^{q+1}(G, A)

for all q ≥ 1.

Remark 17. This shows that the functor H^{q+1}(G, −) is completely determined by H^{i}(G, −) for i ≤ q, giving the claimed uniqueness.

We now explicitly consider H^{q}(G, A) as a quotient of cocycles by coboundaries. We have the following free resolution of ℤ as a G-module

… → P_{2} ⟶^{d_{2}} P_{1} ⟶^{d_{1}} P_{0} ⟶^{d_{0}} ℤ → 0  (†)

where P_{i} = ℤ[G^{i+1}] = ℤ[G × ⋯ × G] with G-action given by s.(g_{0}, …, g_{i}) = (s.g_{0}, …, s.g_{i}) and d: (g_{0}, …, g_{i}) ↦ ∑_{j=0}^{i}(−1)^{j}(g_{0}, …, ĝ_{j}, …, g_{i}).

Lemma 18. P_{i} is a free G-module.

Proof. The set {(1, g_{1}, g_{1}g_{2}, …, g_{1} ⋯ g_{i}) : g_{1}, …, g_{i} ∈ G} is a basis for P_{i} over ℤ[G].

Proposition 19. The free resolution (†) is exact.

Proof. We first show that d_{i} ∘ d_{i+1} = 0. This is a standard calculation from the definitions, for example

\begin{aligned}(d_{1}d_{2})(g_{0}, g_{1}, g_{2}) & = d_{1}((g_{1}, g_{2}) − (g_{0}, g_{2}) + (g_{0}, g_{1})) \\ & = ((g_{2}) − (g_{1})) − ((g_{2}) − (g_{0})) + ((g_{1}) − (g_{0})) = 0. \end{aligned}

Thus, Im(d_{i+1}) ⊂ ker(d_{i}). To prove equality, fix s ∈ G and define a family of maps h: P_{i} → P_{i+1}, h(g_{0}, …, g_{i}) ↦ (s, g_{0}, …, g_{i}). We claim that dh + hd = 1,

(dh + hd)(g_{0}, …, g_{i}) = (dh)(g_{0}, …, g_{i}) + hd(g_{0}, …, g_{i})

= d(s, g_{0}, …, g_{i}) + ∑_{j=0}^{i}(−1)^{j}(s, g_{0}, …, ĝ_{j}, …, g_{i})

= (g_{0}, …, g_{i}) + ∑_{j=1}^{i+1}(−1)^{j+1}(s, g_{0}, …, ĝ_{j}, …, g_{i})

+ ∑_{j=0}^{i}(−1)^{j}(s, g_{0}, …, ĝ_{j}, …, g_{i})

= 0

Let x ∈ ker(d_{i}). Then dhx+hdx = x, and since hdx = 0 we have d(hxd) = x so x ∈ Im(d_{i+1}).

Recall that Hom_{G}(−, A) is a contravariant functor and let K_{i} = Hom_{G}(P_{i}, A). We obtain a complex

0 → K_{0} ⟶^{d_{0}} K_{1} ⟶^{d_{1}} K_{2} ⟶^{d_{2}} K_{3} ⟶^{d_{3}} …

such that H^{q}(G, A) = ker(d_{q})/ Im(d_{q−1}).

Suppose that f ∈ Hom_{G}(P_{i}, A) and define the set-theoretic map ϕ: G^{i} → A by ϕ(g_{1}, …, g_{i}) = f(1, g_{1}, g_{1}g_{2}, …, g_{1} ⋯ g_{i}).

Proposition 20.

\begin{aligned}(dϕ)(g_{1}, …, g_{i+1}) & = g_{1}ϕ(g_{2}, g_{3}, …, g_{i+1}) + ∑_{j=1}^{i}(−1)^{j}ϕ(g_{1}, g_{2}, …, g_{j}g_{j+1}, …, g_{i+1}) \\ & + (−1)^{i+1}ϕ(g_{1}, …, g_{i}) \end{aligned}

Proof. The map d: P_{i+1} → P_{i} induces a map Hom_{G}(P_{i}, A) → Hom_{G}(P_{i+1}, A) via ϕ ↦ ϕ ∘ d,

(dϕ)(g_{1}, …, g_{n}) = f(d(1, g_{1}, g_{1}g_{2}, …, g_{1} ⋯ g_{i+1}))

= f((g_{1}, g_{1}g_{2}, …, g_{1} ⋯ g_{i+1})

+ ∑_{j=1}^{i}(−1)^{j}(1, g_{1}, …, { \atop g_{1} ⋯ g_{j}}, …, g_{1} ⋯ g_{i+1})

+ (−1)^{i+1}(1, g_{1}, g_{1}g_{2}, …, g_{1} ⋯ g_{i}))

= g_{1}ϕ(g_{2}, g_{3}, …, g_{i+1}) + ∑_{j=1}^{i}(−1)^{j}ϕ(g_{1}, …, g_{j}g_{j+1}, …, g_{i+1})

+ (−1)^{i+1}ϕ(g_{1}, …, g_{i})

We now make some explicit considerations in small cases. Begin with the case i = 0. If ϕ ∈ Hom_{G}(P_{0}, A) then there is an a ∈ A such that ϕ() = a. Thus, (dϕ)(g) = gϕ() − ϕ() = ga − a. Now let i = 1 and take ϕ ∈ Hom_{G}(P_{1}, A). Then

(dϕ)(g_{1}, g_{2}) = g_{1}ϕ(g_{2}) − ϕ(g_{1}g_{2}) + ϕ(g_{1})

Thus

H^{0}(G, A) = A^{G}
H^{1}(G, A) = {ϕ: G → A : dϕ = 0}/{dϕ : ϕ ∈ Hom_{G}(P_{0}, A)}
= {ϕ: G → A : ϕ(g_{1}g_{2}) = ϕ(g_{1}) + g_{1}ϕ(g_{2})}/{ϕ_{a}: G → A : a ∈ A}

where ϕ_{a}(g) = ga − a,

H^{2}(G, A) = {ϕ: G × G → A : g_{1}ϕ(g_{2}, g_{3}) − ϕ(g_{1}g_{2}, g_{3})
+ ϕ(g_{1}, g_{2}g_{3}) − ϕ(g_{1}, g_{2}) = 0}/⋆

5 Morphisms of pairs

5.1 Morphism of pairs

A morphism of pairs (G, A) → (G′, A′) together with homomorphisms f: G′ → G and g: A → A′ such that g(f(s′).a) = s′.g(a) for all s′ ∈ G′, a ∈ A induces morphisms H^{q}(G, A) → H^{q}(G′, A′) for all q ≥ 0.

We now explain this in detail. f: G′ → G induces a morphism of the resolutions of ℤ in Mod_{G} and Mod_{G′}.

\xymatrix{⋯ \ar[r] & ℤ[G′ × G′] \ar[r] \ar[d] & ℤ[G′] \ar[r] \ar[d]_{f} & ℤ \ar[r] & 0 \\ ⋯ \ar[r] & ℤ[G × G] \ar[r] & ℤ[G] \ar[r] & ℤ \ar[r] & 0}

Hence we obtain a morphism of corresponding complexes, in the other direction:

\xymatrix{0 \ar[r] & Hom_{G}(ℤ[G], A) \ar[d] \ar[r] & Hom_{G}(ℤ[G × G], A) \ar[d] \ar[r] & ⋯ \\ 0 \ar[r] & Hom_{G′}(ℤ[G′], A) \ar[r] & Hom_{G′}(ℤ[G′ × G′], A) \ar[r] & ⋯}

where the map Hom_{G}(ℤ[G], A) → Hom_{G′}(ℤ[G′], A) is given by (ϕ: ℤ[G] → A) ↦ (g ∘ ϕ ∘ f: ℤ[G′] → A′). Thus, a morphism of pairs (G, A) → (G′, A′) induces a morphism on cohomology H^{q}(G, A) → H^{q}(G′, A′).

5.2 Shapiro’s lemma

We begin by considering a special case. Let G′ ≤ G be a subgroup and A′ ∈ Mod_{G′}. Then form a left G-module A = Hom_{G′}(ℤ[G], A′) by coinduction from A′.

Given a G-action ϕ: ℤ[G] → A′ we set, for s ∈ G, s.ϕ: t ↦ ϕ(ts). For fixed s ∈ G, the map ψ_{s}: ℤ[G] → ℤ[G] given by t ↦ ts is a left G-module homomorphism; ϕ ∘ ψ_{s} would thus give a right G-action, so s.ϕ as above is a left action.

Theorem 21 (Shapiro). For all q ≥ 0,

H^{q}(G, A) ≅ H^{q}(G′, A′).

Remark 22. This is very important in making Eichler–Shimura explicit and computable.

Proof. If L^{•} ↠ ℤ is a free ℤ[G]-resolution, then L^{′•} ↠ ℤ is also a free ℤ[G′]-resolution, since G′ ≤ G is a subgroup. We have an isomorphism

Hom_{G}((L^{i}, Hom_{G′}(ℤ, A′)) = Hom_{G}(L^{i}, A) ≅ Hom_{G′}(L^{i}, A′)

given by ψ ↦ ψ(b)(1) with inverse ϕ ↦ (b ↦ (s ↦ ϕ(bs))). We conclude that (G, A) and (G′, A′) have the same complexes and thus the same cohomology.

6 Inflation and restriction

Suppose that H is a subgroup of G and let A ∈ Mod_{G}. A morphism of pairs (G, A) → (H, A) by H → G and id: A → A induces a restriction homomorphism

res_{H}: H^{q}(G, A) → H^{q}(H, A).

For example, in terms of 1-cocyles, we see that the homomorphism res_{H}: H^{1}(G, A) → H^{1}(H, A), [f] ↦ [f|_{H}] is given by the usual restriction map.

Next suppose that H ◁ G. We obtain a morphism of pairs (G/H, A^{H}) → (G, A). This induces an inflation homomorphism

inf : H^{q}(G/H, A^{H}) → H^{q}(G, A).

We shall later consider the following theorem, whose full proof uses spectral sequences:

Theorem 23. The sequence

0 → H^{1}(G/H, A^{H}) ⟶^{inf} H^{1}(G, A) ⟶^{res} H^{1}(H, A) → H^{2}(G/H, A^{H}) → …

is exact.

7 Inner automorphisms

Suppose t ∈ G. The conjugation s ↦ tst^{−1} is an inner automorphism of G. This induces a morphism of pairs (G, A) → (G, A): G ← G, tst^{−1} ↤ s, A → A, a ↦ t^{−1}a.

We obtain a homomorphism H^{q}(G, A) → H^{q}(G, A), that is, a natural action of G on H^{q}(G, A).

Proposition 24. This action is trivial, that is, the map t ∈ G induces the identity map on H^{q}(G, A).

Proof. Use dimension shifting. For q = 0, consider

[figure: H^{0}(G, A); Hom_{G}(ℤ[G], A); d; ⋯; Hom_{G}(ℤ[G], A); d; ⋯; H^{0}(ℤ[G], A)]

If we let f: G → G, s ↦ tst^{−1} and g: A → A, a ↦ t^{−1}.a then, for ϕ ∈ ker(d) ⊂ Hom_{G}(ℤ[G], A) so that ϕ(1) ∈ A^{G}, we have that

(g ∘ ϕ ∘ f)(1) = g(ϕ(t1t^{−1})) = t^{−1}.ϕ(1) = ϕ(1).

Thus, the induced map on H^{0}(G, A) is the identity.

Now let q ≥ 1 and consider the short exact sequence 0 → A → A^{∗} → A′ → 0 with A^{∗} co-induced. We have the following functorial isomorphisms:

H^{1}(G, A) ≅ coker(H^{0}(G, A^{∗}) → H^{0}(G, A′)),
H^{q}(G, A) ≅ H^{q−1}(G, A′), for q ≥ 2.

By induction, H^{q−1}(G, A′) ≅ H^{q−1}(G, A′) via conjugation by t, so the proposition follows.

8 An application of inner automorphisms

Let k be a field with char(k) ≠ 2. Let G = GL_{n}(k) and A = k^{n}, equipped with the natural G-action.

Proposition 25. H^{q}(G, A) = 0 for all q ≥ 0.

Proof. Let

t = (\begin{matrix}−1 & 0 \\ 0 & −1 \end{matrix}) ∈ G.

Then we proved above that (G, A) → (G, A) induced by conjugation by t induces the identity map H^{q}(G, A) → H^{q}(G, A).

But in fact it induces multiplication by −1 on A:

[figure: s = t^{−1}st; s; G; G; A; A; a; t^{−1}a = −a]

Thus it induces the map H^{q}(G, A) ⟶^{−1} H^{q}(G, A), and we conclude that, on the k-vector space H^{q}(G, A), we have −1 = 1 and so H^{q}(G, A) = 0 for all q ≥ 0.

Remark 26. In the proof we can replace G by any subgroup of GL_{n}(k) that contains a non-identity scalar t.

Example 27. Let E/k be an ellipctic curve and suppose that ρ̄_{E,p}: G_{k} ↠ Aut(E[p]) is surjective. Then

H^{q}(Gal(k(E[p])/k), E[p]) = 0

for all q ≥ 0.

9 The restriction-inflation sequence

Proposition 28. Let H ◁ G be a normal subgroup and A ∈ Mod_{G}. Then we have an exact sequence

0 → H^{1}(G/H, A^{H}) ⟶^{inf} H^{1}(G, A) ⟶^{res} H^{1}(H, A)

Proof. We check this directly on 1-cocycles, following Atiyah–Wall. We begin with exactness at H^{1}(G/H, A^{H}), that is, the injectivity of inf. Consider

H^{1}(G/H, A^{H}) ⟶^{inf} H^{1}(G, A), [f] ↦ [f̄] where

\xymatrix{G \ar@{->>}[dr] \ar[rr]^{f̄} &  & A \\  & G/H \ar[r]^{f} & A^{H} \ar@{^{(}->}[u]}

and suppose that [f̄] = 0. Thus, f̄ is a coboundary and so there exists an a ∈ A such that f̄(s) = s.a − a for all s ∈ G. Note that f̄ is constant on each coset of H in G, and so, for all t ∈ H,

s.a − a = f̄(s) = f̄(st) = (st).a − a

hence s.a = (st).a and a = t.a, which implies a ∈ A^{H}. Thus, f is also a coboundary (attached to a ∈ A^{H}), and hence [f] = 0.

Next we show that res ∘ inf = 0, i.e., Im(inf) ⊂ ker(res). Consider the composition

H^{1}(G/H, A^{H}) ⟶^{inf} H^{1}(G, A) ⟶^{res} H^{1}(H, A), [f] ↦ [f̄] ↦ [f̄|_{H}].

But f̄ is constant on cosets of H, so f̄|_{H}(t) = f̄|_{H}(1) = 0 since f(1.1) = f(1) + 1.f(1) = f(1) + f(1).

Finally, we prove the exactness at H^{1}(G, A), i.e., iker(res) ⊂ Im(inf). Consider an f: G → A such that

H^{1}(G, A) → H^{1}(H, A), [f] ↦ [0].

Then there exists an a ∈ A such that, for all t ∈ H, f(t) = t.a−a. It follows that [f] = [f − ϕ_{a}] where ϕ_{a} is defined by ϕ_{a}(s) = s.a − a, for all s ∈ G. Thus, we may assume that f|_{H} = 0.

If t ∈ H, then

f(st) = f(s) + s.f(t) = f(s)

so f is constant on cosets of H in G. Thus, f defines a function f̃: G/H → A. For s ∈ H, t ∈ G, we have

f(t) = f(st) = 0 + s.f(t)

and hence f(t) ∈ A^{H} and so the image of f̃ lies in A^{H}. Thus, [f̃] ↦ [f], completing the proof.

10 Cohomology sets

For this section, we refer to pp. 123–126 of Serré’s Local Fields. As before, we let G be a group and A a group with a G-action. But A is no longer assumed to be abelian.

For example, we could consider G = Gal(k̄/k) and A the points of any algebraic group over k, e.g., GL_{n}(k̄) or SL_{n}(k̄).

Definition 29. We set H^{0}(G, A) = A^{G} = {a ∈ A : s.a = a for all s ∈ G}, which is a subgroup of A. We define H^{1}(G, A) as 1-cocycles modulo an equivalence relation. Here, a 1-cocycle is a set-theoretic map G → A, s ↦ a_{s} such that a_{st} = a_{s}.s(a_{t}), and we say a_{s} ∼ b_{s} if there exists an a ∈ A with b_{s} = a^{−1}.a_{s}.s(a) for all s ∈ G.

Remark 30. H^{1}(G, A) is a pointed set, with distinguished element the image of the 1-cocycle a_{s} = 1, i.e., the map sending the all of G to 1 ∈ A.

We have described two functors: H^{0}(G, −) is a functor from the category of non-abelian G-modules to the category of groups, and H^{1}(G, −) is a functor from the same category to the category of pointed sets. For example, this means that a G-morphism A → B induces morphisms H^{0}(G, A) → H^{0}(G, B) and H^{1}(G, A) → H^{1}(G, B).

We now explain how the long exact sequence can be salvaged in the context of non-abelian G-modules. Suppose we have a short exact sequence

1 → A ⟶^{ι} B ⟶^{ρ} C → 1  (†)

of non-abelian G-modules. We want to show the existence of a connecting morphism δ: C^{G} → H^{1}(G, A), c ↦ [a_{s}].

Let c ∈ C^{G} and b ∈ B such that ρ(b) = c. Then s(b) ≡ b modulo ι(A) for all s ∈ G, since

ρ(s(b).b^{−1}) = s(c).c^{−1} = cc^{−1} = 1.

So let a_{s} = ι^{−1}(b^{−1}.s(b)) ∈ A. We claim that a_{s} is a well-defined cocycle. In order to simplify notation, let us assume A ⊂ B. Then

a_{st} = b^{−1}(st)(b) = b^{−1}s(b)s(b^{−1}t(b)) = a_{s}.s(a_{t}),

showing that a_{s} is a cocycle. Moreover, if ρ(b′) = ρ(b) = c then b′ = ba for some a ∈ A by exactness and hence

a′_{s} = a^{−1}b^{−1}s(b)s(a) = a^{−1}a_{s}s(a) ∼ a_{s},

which shows that a_{s} is well-defined.

If we suppose that A lies in the centre of B, so that A is abelian, we can define H^{2}(G, A) as before and also obtain a morphism Δ: H^{1}(G, C) → H^{2}(G, A).

Proposition 31. (i) Let 1 → A ⟶^{ι} B ⟶^{ρ}→ 1 be any exact sequence of non-abelian G-modules. Then the sequence

1 → H^{0}(G, A) ⟶^{ι_{0}} H^{0}(G, B) ⟶^{ρ_{0}} H^{0}(G, C)

⟶^{δ} H^{1}(G, A) ⟶^{ι_{1}} H^{1}(G, B) ⟶^{ρ_{1}} H^{1}(G, C)

is exact.

(ii) If, moreover, A lies in the centre of B then the above exact sequence can be extended by H^{1}(G, C) ⟶^{Δ} H^{(}G, A) to another exact sequence.

Proof. See book. (Serre??)

11 Long exact sequence of cohomology sets

This section is motivated by the following example. Let C be a curve defined over a field k with non-abelian automorphism group Aut(C). Then H^{0}(Gal(k̄, k), Aut(C)) is isomorphic to the twists of C over k.

Consider a map π: X → Y of pointed sets with distinguished elements x and y. By definition, ker(π) = π^{−1}(y) ⊂ X.

We first revisit the connecting homomorphism δ. Suppose that 1 → A ⟶^{ι} B ⟶^{ρ} C → 1 is an exact sequence of non-abelian G-modules. The connecting homomorphism is a map δ: H^{0}(G, C) → H^{1}(G, A).

Proposition 32. The sequence

1 → H^{0}(G, A) ⟶^{ι_{0}} H^{0}(G, B) ⟶^{ρ_{0}} H^{0}(G, C)

⟶^{δ} H^{1}(G, A) ⟶^{ι_{1}} H^{1}(G, B) ⟶^{ρ_{1}} H^{1}(G, C)

is exact.

Proof. We continue to identify A ⊂ B. Firstly, the sequence is exact at H^{0}(G, A) since ι_{0} is injective since A^{G} → B^{G}. For exactness at H^{0}(G, B), note that (ρ ∘ ι)(a) = 0 for all a ∈ A, so the same is true on A^{G} and B^{G}. Furthermore, if b ∈ ker(ρ_{0}) ⊂ B^{G} then b ∈ ker(ρ) = Im(ι) = A, so b ∈ A ∩ B^{G} = A^{G} and hence b ∈ Im(ι_{0}), showing that ker(ρ_{0}) ⊂ Im(ι_{0}).

Next, we prove exactness at H^{0}(G, C). We can describe ρ_{0}(B^{G}) as the set of c ∈ C^{G} that lift to invariant elements of B, and ker(δ) as the set of c ∈ C^{G} such that a_{s} = b^{−1}s(b) ∼ 1 in H^{1}(G, A) whenever b ↦ c. Suppose that c ∈ ker(δ) so that c ∈ C^{G} and there exists b ∈ B^{G} such that b ↦ c and a_{s} = b^{−1}s(b) ∼ 1. Then a_{s} is trivial if and only if a_{s} ∼ 1, i.e., there exists an a ∈ A with

a^{−1}b^{−1}.s(b)s(a) = 1

for all s ∈ G, that is, s(ba) = ba and thus ba ∈ B^{G}.

Then a_{s} ∼ (ba)^{−1}s(ba) and ba ∈ B^{G}. But a ∈ A so ρ(ba) = ρ(b) = c, so ba ↦ c and ba ∈ B^{G} hence c ∈ ρ_{0}(B^{G}).

Now, we prove exactness at H^{1}(G, A). Suppose that ι_{1} sends [a_{s}] to the trivial element of H^{1}(G, B). Then there exists a b ∈ B such that a_{s} ∼ b^{−1}s(b) for all s ∈ G. Thus [a_{s}] = δ(ρ(b)), which shows that ker(ι_{1}) ⊂ Im(δ). For the other direction, note that δ(c) is given by a_{s} = b^{−1}s(b) for some b such that b ↦ c. But this means that ι_{1}(a_{s}) ∼ 1.

Finally, we prove exactness H^{1}(G, B). First, note that a_{s} ∈ H^{1}(G, A) maps to s ↦ ρ_{1}(ι_{1}(a_{s})) = 1, since rho∘ι = 1, showing that Im(ι_{1}) ⊂ ker(ρ_{1}). For the other direction, let b_{s} ∈ H^{1}(G, B) such that (s ↦ ρ_{1}(b_{s})) ∼ 1. Then there exists a c ∈ C such that ρ_{1}(b_{s}) = c^{−1}s(c). We may modify by a lift of c so that ρ_{1}(b_{s}) = 1 for all s. It then follows that b_{s} ∈ A for all s, so that [b_{s}] ∈ Im(ι_{1}).

12 Homology

In this section, we let G be a group, A an abelian G-module and set DA = ⟨s.a − a : a ∈ A, s ∈ G⟩, a subgroup of A.

We first observe that DA is in fact a G-module,

t.(s.a − a) = t.s.a − t.a = (tst^{−1}).(t.a) − t.a ∈ DA.

Thus, A/DA is also a G-module, with trivial G-action.

Definition 33. We define H_{0}(G, A) = A/DA, which is the largest quotient of A with trivial G-action. For q ≥ 1, we also let

H_{q}(G, A) = Tor_{q}^{ℤ[G]}(ℤ, A)

so that H_{q}(G, −) are the left-derived functors of H_{0}(G, −).

Given a short exact sequence 0 → A → B → C → 0, as before we obtain a long exact sequence

⋯ → H_{1}(G, A) → H_{1}(G, B) → H_{1}(G, C)

⟶^{δ} H_{0}(G, A) → H_{0}(G, B) → H_{0}(G, C) → 0.

Example 34. For any group G, we have H_{1}(G, ℤ) = G/G′, the abelianisation of G, where G′ = ⟨aba^{−1}b^{−1} : a, b ∈ G⟩ is the commutator subgroup. See Serre §VII.4 for a proof.

13 Tate cohomology groups

The Tate cohomology groups naturally relate homology and cohomology, and make several results much more natural.

Suppose that G is a finite group and let A be a G-module. We define the norm element, linking homology and cohomology, by

N = ∑_{s∈G} s.

This induces a map N: A → A, a ↦ N(a) = ∑_{s∈G} s.a, which is in fact a G-module homomorphism, since

t ∑_{s∈G} s.a = ∑_{s∈G}(ts).a = ∑_{s∈G}(st).a,

noting that st and ts range over the same elements of G for s ∈ G. We define the augmentation ideal as

I_{G} = (s − 1 : s ∈ G) ⊂ ℤ[G].

Lemma 35. (i) I_{G}A ⊂ ker(N), which we will also denote A[N].

(ii) Im(N) ⊂ A^{G}.

Proof. We use that, for fixed s ∈ G and variable t ∈ G, st ranges over all elements of G as t does:

(i)

N((s − 1).a) = (∑_{t∈G} t).(s − 1).a = (∑_{t∈G} ts − ∑_{t∈G} t).a = 0.

(ii)

s.N(a) = s. ∑_{t∈G} t.a = ∑_{t∈G}(st).a = ∑_{t∈G} t.a = N(a).

This induces the aforementioned link between homology and cohomology as follows:

[figure: A; N; A; H_{0}(G, A); N^{∗}; H^{0}(G, A); Ĥ_{0}(G, A) = ker(N^{∗}); Ĥ^{0}(G, A) = coker(N^{∗}); A[N]/I_{G}A; 0; A^{G}/N(A)]

Proposition 36. If A is induced then Ĥ^{0}(G, A) = 0.

Proof. The assumption that A is induced means that A ≅ ⨁_{s∈G} s.X ≅ ℤ[G] ⊗_{ℤ} X, where X ≤ A is a subgroup. This is easy to see since G is finite.

Thus, each a ∈ A can be expressed uniquely as

a = ∑_{s∈G} s.x_{s}, x_{s} ∈ X

Now a ∈ A^{G} if and only if all x_{s} are equal if and only if a = N(x_{s}). This shows that A^{G} = N(A), as claimed.

Remark 37. If A is induced, we also have that Ĥ_{0}(G, A) = 0. The proof of this statement will appear on the problem sheet.

Definition 38. We define the Tate cohomology groups as follows:

Ĥ^{q}(G, A) = H^{q}(G, A), q ≥ 1
Ĥ^{0}(G, A) = A^{G}/N(A)
Ĥ^{−1}(G, A) = A[N]/I_{G}A
Ĥ^{−q}(G, A) = H_{q−1}(G, A), q ≥ 2

Remark 39. The functor Ĥ^{0}(G, −) is not left exact.

Proposition 40. Given a short exact sequence of G-modules 0 → A → B → C → 0, we have the long exact sequence

⋯ → Ĥ^{−2}(G, C) → Ĥ^{−1}(G, A) → Ĥ^{−1}(G, B) → Ĥ^{−1}(G, C)

→ Ĥ^{0}(G, A) → Ĥ^{0}(G, B) → Ĥ^{0}(G, C)

→ Ĥ^{1}(G, A) → Ĥ^{1}(G, B) → Ĥ^{1}(G, C) → Ĥ^{2}(G, A) → ⋯ Proof. We have the following commutative diagram of exact sequences:

\xymatrix{Ĥ^{−2}(G, C) \ar[d]_{≅} & Ĥ^{−1}(G, A) \ar[r] \ar@{^{(}->}[d] & Ĥ^{−1}(G, B) \ar[r] \ar@{^{(}->}[d] & Ĥ^{−1}(G, C) \ar@{^{(}->}[d] \\ H_{1}(G, C) \ar[d] \ar[r] \ar@{-->}[ur] & H_{0}(G, A) \ar[d]^{N_{A}^{∗}} \ar[r] & H_{0}(G, B) \ar[d]^{N_{B}^{∗}} \ar[r] & H_{0}(G, C) \ar[d]^{N_{C}^{∗}} \ar[r] & 0 \\ 0 \ar[r] & H^{0}(G, A) \ar@{->>}[d] \ar[r] & H^{0}(G, B) \ar@{->>}[d] \ar[r] & H^{0}(G, C) \ar@{->>}[d] \ar[r] & H^{1}(G, A) \\  & Ĥ^{0}(G, A) \ar[r] & Ĥ^{0}(G, B) \ar[r] & Ĥ^{0}(G, C) \ar@{-->}[ur]}

which yields a connecting homomorphism Ĥ^{−1}(G, C) → Ĥ^{0}(G, A) by the Snake Lemma.

Fact 41. (i) Every G-module A embeds in a G-module A^{∗} such that

Ĥ^{q}(G, A^{∗}) = 0

for all q ∈ ℤ.

(ii) Every G-module A is a quotient of some G-module A_{∗} such that

Ĥ^{q}(G, A_{∗}) = 0

for all q ∈ ℤ.

This is extremely useful, see p. 129 of Serre.

We continue with the assumption that G is a finite group and that A is a G-module. In this lecture, we will obtain the following results, each valid for every q ∈ ℤ:

(i) Ĥ^{q}(G, A) is killed by |G|;

(ii) if A is finitely generated then Ĥ^{q}(G, A) is finite;

(iii) if S ≤ G is a Sylow p-subgroup, then we have an injection

res: Ĥ^{q}(G, A)(p) → Ĥ^{q}(S, A),

where Ĥ^{q}(G, A)(p) denotes the p-primary subgroup.

So far we do not have the tools to prove these facts!

14 Complete resolution of G

Recall our earlier definition of P_{•}, giving a free ℤ[G]-resolution of ℤ. Dualising this by defining P_{i}^{∗} = Hom_{ℤ}(P_{i}, ℤ) over ℤ, not ℤ[G], we obtain a second exact sequence

[figure: ⋯; P_{2}; P_{1}; P_{0}; ε; ℤ; 0; 0; ℤ; ε^{∗}; P_{0}^{∗}; P_{1}^{∗}; P_{2}^{∗}; ⋯]

We extend this to negative indices by setting P_{−n} = P_{n−1}^{∗}. Thus, we obtain a long exact sequence

… → P_{2} → P_{1} → P_{0} → P_{−1} → P_{−2} → …

called a complete resolution.

Proposition 42. Ĥ^{q}(G, A) is the q-th cohomology group of the complex and we have an exact sequence

⋯ ← Hom(P_{1}, A) ← Hom(P_{0}, A) ← Hom(P_{−1}, A) ← Hom(P_{−2}, A) ← · · · .

Proof. In the case q ≥ 1, this follows immediately from the definition.

Suppose now that q ≤ −2. By definition, H_{n}(G, A) is the nth homology of the complex P_{•}⊗_{ℤ[G]}A, i.e., the standard complex, although any projective resolution of ℤ will work. But for a finitely generated free ℤ[G]-module B,

B × A ⟶^{∼} Hom(B^{∗}, A), c × a ↦ (f ↦ f(c).a),

where B^{∗} = Hom(B, ℤ), is an isomorphism of ℤ[G]-modules. So we obtain an isomorphism τ

τ: B ⊗_{ℤ[G]} A = (B ⊗ A)/I_{G}.(B ⊗ A) ⟶_{N^{∗}}^{∼} (B ⊗ A)^{G}

→ Hom(B^{∗}, A)^{G} = Hom_{G}(B^{∗}, A),

which can be verified using the identities b⊗s.a = s^{−1}.b⊗a, s.b⊗s.a = b⊗a, and (s − 1).(b ⊗ a) = s.b ⊗ s.a − b ⊗ a. Moreover, we note that N^{∗} is an isomorphism since B ⊗_{ℤ} A is induced, which is the case because B is free and the tensor product is formed over ℤ. Thus

Hom_{G}(P_{−n}, A) = Hom_{G}(Hom(P_{n−1}, ℤ), A) ≅ P_{n−1} ⊗_{ℤ[G]} A

and the result follows since we are computing cohomology of the same complex.

For the cases q = 0 and q = 1, see Cassels–Fröhlich, p. 103.

Recall the technique of dimension shifting: given an exact sequence 0 → A′ → A_{∗} → A → 0, where A_{∗} is an induced G-module, say of the form ℤ[G] ⊗_{ℤ} A, then, for all q ∈ ℤ,

Ĥ^{q}(G, A) ⟶_{δ}^{∼} Ĥ^{q+1}(G, A′).

Taking H ≤ G allows us to define the restriction map for all q ∈ ℤ via the following commutative diagram

\xymatrix{Ĥ^{q}(G, A′) \ar[r]^{res} \ar[d]_{≅} & Ĥ^{q}(H, A′) \\ Ĥ^{q−1}(G, A) \ar[r]^{res} & Ĥ^{q−1}(H, A)}

15 Corestriction

For homology, a morphism of pairs

[figure: G′; G; A′; A]

induces a homomorphism H_{q}(G′, A′) → H_{q}(G, A).

Thus, for H → G we obtain a homomorphism H_{q}(H, A) → H_{q}(G, A) for all q. It follows that, for an exact sequence 0 → A → A^{∗} → A′ → 0 as above, we have the following commutative diagram for all q ∈ Z

\xymatrix{Ĥ^{q}(G, A) \ar[r]^{∼} & Ĥ^{q−1}(G, A′) \\ Ĥ^{q}(H, A) \ar[u]_{cores} \ar[r]^{∼} & Ĥ^{q−1}(H, A′) \ar[u]_{cores}}

Proposition 43. • The homomorphism res: Ĥ^{0}(G, A) → Ĥ^{0}(H, A) is given by the homomorphism A^{G}/N(A) → A^{H}/N(A), induced by the inclusion A^{G} → A^{H}.

• The homomorphism res: Ĥ^{−1}(G, A) → Ĥ^{−1}(H, A) is induced by the homomorphism N′_{G/H}: A/I_{G}(A) → A/I_{H}(A) sending the image of a to ∑ s_{i}^{−1}.a, where the s_{i} ∈ G are chosen such that G/H = ⋃ s_{i}.H.

• The homomorphism cores: Ĥ^{0}(H, A) → Ĥ^{0}(G, A) is induced by the homomorphism N_{G/H}: A^{H}/N_{H}(A) → A^{G}/N_{G}(A) where N_{G/H}(a) = ∑ s_{i}.a.

Proof. See Cassels–Fröhlich. Proposition 44. For all q ∈ ℤ,

cores_{G/H} ∘ res_{H}: Ĥ^{q}(G, A) → Ĥ^{q}(G, A)

and the composition cores ∘ res is multiplication by [G : H]. Proof. We check this for q = 0; the general result will then follow by dimension shifting.

\xymatrix{Ĥ^{0}(G, A) = A^{G}/N(A) \ar[d]_{res} \\ Ĥ^{0}(H, A) = A^{H}/N(A) \ar[u]_{cores}}

where the restriction map is induced by A^{G} ⊂ A^{H} and the co-restriction map is induced by N_{G/H}(a) = ∑ s_{i}.a, where the s_{i} are coset representatives of H in G. Observe that, for a ∈ A^{G},

cores(res(a)) = [∑ s_{i}.a] = n.a

where n = [G : H]. Corollary 45. (i) Ĥ^{q}(G, A) is killed by |G|, for all q ∈ ℤ. (ii) If A is finitely generated then Ĥ^{q}(G, A) is finite for all q ∈ ℤ. (iii) Suppose S ≤ G is a Sylow p-subgroup, that is, |S| = p^{n} | |G| but

p^{n+1} ∤ |G|. Then Ĥ^{q}(G, A)(p) → Ĥ^{q}(S, A), for all q ∈ ℤ. Proof. (i) Take H = {1} ≤ G above and note that Ĥ^{q}(H, A) = 0 for all

q ∈ ℤ. (ii) This is a calculation of Ĥ^{q}(G, A) using the standard resolution. One

shows that Ĥ^{q}(G, A) is finitely generated, but since every element is killed by |G|, it follows that it is finite.

(iii) We use the composition of maps

\xymatrix{Ĥ^{q}(G, A) \ar[r]_{res} & Ĥ^{q}(S, A) \ar[l]_{cores}}

Suppose |G| = p^{n}m and let x ∈ Ĥ^{q}(G, A) be an element of order a power of p. Then (cores ∘ res)(x) = 0 = m · x, thus x = 0 and res(x) = 0.

16 Cup Product

We will define and construct the cup product pairing on Tate cohomology groups and describe some of its basic properties. The main references are §7 of Atiyah-Wall, §VIII.3 of Serre’s Local Fields, Washington’s paper Galois Cohomology (in Cornell-Silverman-Stevens), and §7 of Tate’s Galois Cohomology (PCMI). The cusp product is absolutely central to Galois cohomology, in that many of the central theorems and constructions involve various types of duality results, which involve cup products at their core.

16.1 The Definition

Let G be a finite group.

Theorem 16.1. There is a unique family of “cup product” homomorphisms

Ĥ^{p}(G, A) ⊗ Ĥ^{q}(G, B) → Ĥ^{p+q}(G, A ⊗ B)
a ⊗ b ↦ a ∪ b,

for all p, q ∈ ℤ and G-modules A, B, such that:

(i) Cup product is functorial in A, B, e.g., if A → A′, B → B′ are Gmodule homomorphisms, then we have a commutative diagram (with vertical maps that I have not typeset below):

Ĥ^{p}(G, A) ⊗ Ĥ^{q}(G, B) → Ĥ^{p+q}(G, A ⊗ B)
Ĥ^{p}(G, A′) ⊗ Ĥ^{q}(G, B′) → Ĥ^{p+q}(G, A′ ⊗ B′)

(ii) When p = q = 0 the cup product is induced by the natural map

A^{G} ⊗ B^{G} → (A ⊗ B)^{G}.

(iii) A natural compatibility statement that allows for dimension shifting and ensure uniqueness (see Cassels-Frohlich or Serre for the exact statement).

16.2 Existence

Let P_{n} be a complete resolution of G, e.g., P_{n} could be the standard resolution:

P_{n} = \begin{cases}ℤ[G^{n+1}] & if n ≥ 0, \\ Hom(P_{|n|−1}, ℤ) & if n < 0. \end{cases}

Recall that this fit together to form an exact sequence of free G-modules:

⋯ ⟶^{d} P_{2} ⟶^{d} P_{1} ⟶^{d} P_{0} ⟶^{d} P_{−1} ⟶^{d} P_{−2} ⟶^{d} · · · .

Moreover, we have

Ĥ^{q}(G, A) = H^{q}(Hom_{G}(P_{∗}, A))

is the qth cohomology of the complex Hom_{G}(P_{∗}, A). In particular

Ĥ^{q}(G, A) = \frac{ker(Hom_{G}(P_{q}, A) → Hom_{G}(P_{q+1}, A))}{im(Hom_{G}(P_{q−1}, A) → Hom_{G}(P_{q}, A))}.

To prove that the family of cup product morphisms exist, we will construct a G-module homomorphism from the complete resolution with certain properties.

Proposition 16.2. There exist G-module homorphisms

φ_{p,q} : P_{p+q} → P_{p} ⊗ Q_{q}

for all p, q ∈ ℤ such that

(i) φ_{p,q} ∘ d = (d ⊗ 1) ∘ φ_{p+1,q} + (−1)^{p}(1 ⊗ d) ∘ φ_{p,q+1}, and

(ii) (ε⊗ε)∘φ_{0,0} = ε, where ε : P_{0} → ℤ is defined by ε(g) = 1 for all g ∈ G.

Assume that the proposition has been proved. Then we define the cup product explicitly on the level of cochains as follows. Let

f ∈ Hom_{G}(P_{p}, A), g ∈ Hom_{G}(P_{q}, B)

be cochains (so elements of the kernel of d). Define the cochain

f ∪ g ∈ Hom_{G}(P_{p+q}, A ⊗ B)

by

f ∪ g = (f ⊗ g) ∘ φ_{p,q}.

Lemma 16.3. We have

f ∪ g = (df) ∪ g + (−1)^{p}f ∪ (dg).

Corollary 16.4. If f, g are cochains, then:

(i) f ∪ g is a cochain

(ii) f ∪ g only depends on the classes of f and g.

We conclude that we have a well-defined homomorphism

Ĥ^{p}(G, A) ⊗ Ĥ^{q}(G, B) → Ĥ^{p+q}(G, A ⊗ B).

Proposition 16.5. Condition (ii) of Theorem 16.1 is satisfied.

Proof. This uses from Proposition 16.2 that (ε ⊗ ε) ∘ φ_{0,0} = ε.

It remains to construct the maps φ_{p,q}. These maps are constructed in a natural way in terms of the standard complete resolution P_{n} mentioned above, as follows. First note that if q ≥ 1, then P_{−q} = P_{q−1}^{∗} = Hom(P_{q−1}, ℤ) has a ℤ-module basis consisting of all (g_{1}^{∗}, …, g_{q}^{∗}), where (g_{1}^{∗}, …, g_{q}^{∗}) maps (g_{1}, …, g_{q}) ∈ P_{q−1} to 1 ∈ ℤ, and every other basis element of P_{q−1} to 0. The map d : P_{−q} → P_{−q−1} is then

d(g_{1}^{∗}, …, g_{q}^{∗}) = ∑_{s∈G} ∑_{i=0}^{q}(−1)^{i}(g_{1}^{∗}, …, g_{i}^{∗}, s^{∗}, g_{i+1}^{∗}, … g_{q}^{∗}).

and d : P_{0} → P_{−1} is given by d(g_{0}) = ∑_{s∈G} s^{∗}.

If p ≥ 0 and q ≥ 0, then

φ_{p,q}(g_{0}, … g_{p+q}) = (g_{0}, …, g_{p}) ⊗ (g_{p}, … g_{p+q}),

and if p, q ≥ 1 then

φ_{−p,−q}(g_{1}^{∗}, …, g_{p+q}^{∗}) = (g_{1}^{∗}, … g_{p}^{∗}) ⊗ (g_{p+1}^{∗}, …, g_{p+q}^{∗}),

and similar definitions in other cases, when one of p, q is positive and the other is negative. (Again, see Cassels-Frohlich for more details.) The moral of all this is that one can construct the cup product by simply following your nose.

16.3 Properties

Proposition 16.6. The cup product has these properties:

(i) (a ∪ b) ∪ c = a ∪ (b ∪ c)

(ii) res(a ∪ b) = res(a) ∪ res(b)

(iii) cores(a ∪ res(b)) = cores(a) ∪ b.

The above properties are proved by proving them when p = q = 0, then using dimension shifting.

Finally, notice that if A ⊗ B → C is a G-homomorphism, then cup product induces

Ĥ^{p}(G, A) ⊗ Ĥ^{q}(G, B) → Ĥ^{p+q}(G, C).

See Tate’s paper Galois Cohomology for an explicit description of the cup product for p, q ≥ 0 on cocycles, which would make computation of the cup product of classes represented by cocycles explicit.

16.4 Cohomology of a Cyclic Group

Suppose that G = ⟨s⟩ is a finite cyclic group. In this section we give a quick summary of the basic facts about Ĥ^{q}(G, A).

Let K_{i} = ℤ[G] and define maps d : K_{i+1} → K_{i} by multiplication by s−1 if i is even and multiplication by N = ∑_{t∈G} t if i is odd. Then

⋯ ⟶^{d} K_{i} ⟶^{d} K_{i−1} ⟶^{d} K_{i−2} ⟶^{d} ⋯

is a complete resolution of G, since

ker(T) = ℤ[G]^{G} = N(ℤ[G]) = image(N),

and since Ĥ^{0}(G, ℤ[G]) = 0,

ker(N) = I_{G}ℤ[G] = image(T).

Then Hom_{G}(K_{•}, A) is

⋯ ← A ⟵^{N} A ⟵^{T} A ⟵^{N} · · · .

Proposition 16.7. For every integer q we have

Ĥ^{2q}(G, A) ≅ Ĥ^{0}(G, A) = A^{G}/N(A)

and

Ĥ^{2q+1}(G, A) ≅ Ĥ^{−1}(G, A) = ker(N_{A})/I_{G}(A).

If n = #G, then we have

Ĥ^{2}(G, ℤ) ≅ Ĥ^{0}(G, ℤ) ≅ ℤ^{G}/N(ℤ) = ℤ/nℤ.

Theorem 16.8. Cup product by a generator of Ĥ^{2}(G, ℤ) induces an isomorphism

Ĥ^{q}(G, A) ⟶^{≅} Ĥ^{q+2}(G, A)

for all q ∈ ℤ and all G-modules A.

For the proof, see Cassels-Frohlich, Section 8.

17 Galois Cohomology

In this course we have developed a foundation for group cohomology. The goal for the rest of the course (about 15 lectures), is to see some applications of group cohomology to Galois theory and algebraic number theory that are important to understanding contemporary research in number theory. For the rest of the course, our group will always be the Galois G of an extension of fields, and our module A will “arise in nature”, equipped with a natural action of G. The main topics are Galois cohomology of abelian varieties, the Brauer group of a field, local and global duality, and étale (and other) cohomology which generalizes the idea of Galois cohomology to bases other than fields. This part of the course requires more background in number theory.

Excellent references include the many articles with the title Galois Cohomology, such as Tate’s, Washington’s, etc.

17.1 The Definition

Let K be a field, e.g., a number field such as ℚ(\sqrt[3]{2}), a finite field such as 𝔽_{9}, a p-adic field such as ℚ_{11}, or a function such as 𝔽_{7}(t) or ℂ(u, v). Let L/K be a finite separable Galois extension of K with Galois group G = Gal(L/K). For any G-module A, let

H^{q}(L/K, A) = H^{q}(Gal(L/K), A), for q ≥ 0

and

Ĥ^{q}(L/K, A) = Ĥ^{q}(Gal(L/K), A), for all q ∈ ℤ.

We call A a Galois module.

17.2 Infinite Galois extensions

John Tate pioneered the study of H^{q}(L/K, A) when L/K is infinite. When L is infinite, let

H^{q}(L/K, A) = \underrightarrow{lim}_{M} H^{q} (M/K, A^{Gal(L/M)}) ,

where the injective limit is over all finite Galois extensions M of K contained in L, and the maps are the inflation maps. We will often write

A(M) = A^{Gal(L/M)},

motivated by similar notation for the group of rational points on an elliptic curve.

When K ⊂ M ⊂ M′, we have a morphism of pairs

(Gal(M/K), A(M)) → (Gal(M′/K), A(M′)),

given by the natural map Gal(M′/K) → Gal(M/K) and the inclusion A(M) ↪ A(M′), which defines

H^{q}(M/K, A(M)) ⟶^{inf} H^{q}(M′/K, A(M′)).

When q = 1, the inf-res sequence is exact, so all of the maps used to define the above injective limit are injections, and we can think of H^{1}(L/K, A) as simply being the “union” of the groups H^{1}(M/K, A(M)), over all finite Galois M. When q > 1, (presumably) the above inflation maps need not be injective.

Finally, we let

H^{q}(K, A) = H^{q}(K^{sep}/K, A).

With this notation, the inf-res sequence is

0 → H^{1}(M/K, A(M)) ⟶^{inf} H^{1}(K, A) ⟶^{res} H^{1}(M, A).

The correct topology on the group Gal(L/K) is the one for which the open subgroups are the subgroups Gal(L/M) for M any finite Galois extension of K.

Exercise 17.1. Use the axiom of choice to show that there exists a finite index normal subgroup of Gal(ℚ̄/ℚ) that is not open. [Hint: Consider the compositum of infinitely many distinct quadratic extensions of ℚ. Their Galois group is ∏ 𝔽_{2}. The ideal ⊕𝔽_{2} in ∏ 𝔽_{2} contains a maximal ideal I. Consider the inverse image of I in Gal(ℚ̄/ℚ).]

We always equip A with the discrete topology.

Proposition 17.2. Fix topologies on Gal(L/K) and A as above. Then H^{q}(L/K, A) is the group of continuous cocycles modulo coboundaries.

Proof. We first show that if [f] ∈ H^{q}(L/K, A) then f is continuous. By definition we have [f] ∈ H^{q}(M/K, A(M)) for some finite Galois extension M/K. Since the natural restriction map Gal(L/K)^{q} → Gal(M/K)^{q} is continuous and both Gal(M/K) and A(M) are discrete, we conclude that the composite map f is continuous.

Next suppose f : Gal(L/K)^{q} → A is a continuous cocycle. Because A has the discrete topology, the inverse image of 0 ∈ A is an open set. From the cocycle condition, the inverse image of 0 is a subgroup as well. Thus f factors through some finite Galois extension M, Gal(L/K)^{q} → Gal(M/K)^{q}, hence [f] is defined by an element of H^{q}(M/K, A(M)).

17.3 Some Galois Modules

The following are all examples of G = Gal(L/K)-modules:

(i) The groups A = ℤ and A = ℚ/ℤ with the trivial action.

(ii) The additive group A = (L, +) of L.

(iii) The multiplicative group A = L^{∗} of L.

(iv) When n ≥ 2, the non-commutatative G-modules A = GL_{n}(L) and SL_{n}(L).

(v) When L is a number field, the ring of integers O_{L}, the units O_{L}^{∗}, and the class group Cl(O_{L}) are all G-modules.

(vi) For E an elliptic curve defined over K, the group A = E(L) of Lrational points.

(vii) The group B(L), where B ia an abelian variety over K.

(viii) If S is any (commutative) group scheme over K, then A = S(L) is a (commutative) G-module.

(ix) For any integer n and any (commutative) A elsewhere in this list, the group A[n] of elements of order dividing n is a G-module.

(x) The p-adic Tate module Tate_{p}(A) = \underleftarrow{lim} A[p^{n}] associated to an abelian variety A.

17.4 The Additive and Multiplicative Groups of a Field

We recall some basic facts from Galois theory. Suppose L/K is a finite Galois extension of fields, which means that # Aut(L/K) = [L : K], or equivalently, L is the splitting field of a single irreducible separable polynomial f ∈ K[x]. We write Gal(L/K) = Aut(L/K).

Proposition 17.3. Let L/K be any Galois extension of fields. Then for all q ∈ ℤ,

Ĥ^{q}(L/K, L) = 0.

Proof. Without loss, we may assume that L is a finite extension of K, since otherwise, we use the result on each finite subextension and take the limit. Since L/K is finite separable, by the normal basis theorem from Galois theory, there exists α ∈ L such that, letting Kβ denote the 1-dimensional K-vector space spanned by β, we have

L = ⨁_{σ∈Gal(L/K)} Kσ(α) ≅ K ⊗_{ℤ} ℤ[Gal(L/K)].  (17.1)

But then L is induced, from which the conclusion follows.

Proposition 17.4. Let L/K be any Galois extension of fields. Then

H^{1}(L/K, L^{∗}) = 0.

Proof. As above, we may assume that L/K is a finite extension. Suppose f : Gal(L/K) → L^{∗} is a 1-cocycle. For any c ∈ L, consider the sum

b = ∑_{σ∈Gal(L/K)} f(σ)σ(c).

If b = 0 for all c, then the elements of Gal(L/K) are linearly dependent. But in view of Equation (17.1), this would imply that the conjugates of a normal basis element α would generate a field of degree < [L : K], a contradiction. Thus there exists c ∈ L with b ≠ 0. Then for any σ ∈ Gal(L/K), we have

0 ≠ σ(b) = ∑_{τ} σ(f(τ))στ(c) = ∑_{στ} f(σ)^{−1}f(στ)στ(c) = f(σ)^{−1}b,

so f(σ) = bσ(b)^{−1}, hence f is a coboundary.

Theorem 17.5 (Hilbert’s Theorem 90). Suppose Gal(L/K) is finite cyclic, with generator σ. If α ∈ L^{∗} has norm 1, then there exists β ∈ L^{∗} such that α = β/σ(β).

Proof. Recall that when G is a finite cyclic group and A is a G-module, then

H^{1}(G, A) ≅ ker(N_{A})/I_{G}(A).

By Proposition 17.4, we have H^{1}(L/K, L^{∗}) = 0, so the kernel of norm on L^{∗} equals the image of 1 − σ ∈ ℤ[Gal(L/K)]. Thus α, which is in the kernel of the norm, is of the form (1 − σ)β = β/σ(β) for some β. (Note that the group ring is written additively, which is why minus changes to inverse.)

Remark 17.6. Here is an amusing consequence of Theorem 17.5. Let L = ℚ(i) and K = ℚ. Then α = a + bi ∈ ℚ(i) has norm 1 if and only if a^{2} + b^{2} = 1, i.e., (a, b) is a rational point on the unit circle. Theorem 17.5 asserts that there is β = c + di such that

a + bi = \frac{β}{σ(β)} = \frac{c + di}{c − di} = \frac{(c + di)^{2}}{c^{2} + d^{2}} = \frac{c^{2} − d^{2}}{c^{2} + d^{2}} + \frac{2cd}{c^{2} + d^{2}} i.

This recovers the standard parameterization of rational points on the unit circle.

Later we will study the Brauer group of a field K

Br(K) = H^{2}(K, (K^{sep})^{∗}),

which can be very large and subtle.

18 Kummer Theory

18.1 Kummer Theory of Fields

Kummer theory is concerned with classifying the abelian extensions of exponent n of a field K, assuming that K contains the nth roots of unity. It’s a generalization of the correspondence between quadratic extensions of ℚ and non-square squarefree integers.

Let n be a positive integer, and let K be a field of characteristic prime to n. Let L be a separable closure of K. Let μ_{n}(L) denote the set of elements of order dividing n in L.

Lemma 18.1. μ_{n}(L) is a cyclic group of order n.

Proof. The elements of μ_{n}(L) are exactly the roots in L of the polynomial x^{n} − 1. Since n is coprime to the characteristic, all roots of x^{n} − 1 are in L, so μ_{n}(L) has order at least n. But K is a field, so x^{n} − 1 can have at most n roots, so μ_{n}(L) has order n. Any finite subgroup of the multiplicative group of a field is cyclic, so μ_{n}(L) is cyclic.

Consider the exact sequence

1 → μ_{n}(L) → L^{∗} ⟶^{x ↦ x^{n}} L^{∗} → 1

of G_{K} = Gal(L/K)-modules. The associated long exact sequence of Galois cohomology yields

1 → K^{∗}/(K^{∗})^{n} → H^{1}(K, μ_{n}(L)) → H^{1}(K, L^{∗}) → · · · .

We proved that H^{1}(K, L^{∗}) = 0, so we conclude that

K^{∗}/(K^{∗})^{n} ≅ H^{1}(K, μ_{n}(L)),

where the isomorphism is via the δ connecting homomorphism. If α ∈ L^{∗}, we obtain the corresponding element δ(α) ∈ H^{1}(K, μ_{n}(L)) by finding some β ∈ L^{∗} such that β^{n} = α; then the corresponding cocycle is σ ↦ σ(β)/β ∈ μ_{n}(L).

As a special case, consider n = 2 and K = ℚ. Then we have μ_{2}(ℚ̄) = {±1}, on which G_{ℚ} acts trivially. Recall that H^{1}(G, A) = Hom(G, A) when G acts trivially on A. Thus

ℚ^{∗}/(ℚ^{∗})^{2} ≅ Hom(G_{ℚ}, {±1}),

where the homomorphisms are continuous. The set of squarefree integers are representative elements for the left hand side of the above isomorphism. The right hand side is the set of continuous homomorphisms φ : G_{ℚ} → {±1}. To give such a nontrivial homorphism φ is exactly the same as giving a quadratic extension of ℚ. We thus recover—in a conceptual way—the standard bijection between quadratic fields and squarefree integers ≠ 1, which is one of the basic facts one learns in a first algebraic number theory course.

We generalize the above construction as follows. Suppose μ_{n} ⊂ K, i.e., all the nth roots of unity are already in K. Then we have

K^{∗}/(K^{∗})^{n} ≅ Hom(G_{K}, ℤ/nℤ),  (18.1)

where as usual the homomorphisms are continuous. We associate to a homomorphism φ : G_{K} → ℤ/nℤ an extension L^{H} of K, where H = ker(φ), and by Galois theory, Gal(L^{H}/K) ≅ image(φ) ⊂ ℤ/nℤ. Conversely, given any Galois extension M/K with Galois group contained in ℤ/nℤ, there is an associated homorphism φ : G_{K} → Gal(M/K) ⊂ ℤ/nℤ. Define an equivalence relation ∼ on Hom(G_{K}, ℤ/nℤ) by φ ∼ ψ if ker(φ) = ker(ψ) (equivalently, φ = mψ for some integer m coprime to n). Then we have a bijection

Hom(G_{K}, ℤ/nℤ)_{/∼} ⟶^{≅} { Galois extensions M/K with Gal(M/K) ⊂ ℤ/nℤ }.

Using Equation 18.1 along with the explicit description of δ mentioned above, we thus see that the Galois extensions of K with Gal(M/K) ⊂ ℤ/nℤ are the extensions of the form K( \sqrt[n]{α}) for some α ∈ K^{∗}. An element σ ∈ Gal(M/K) acts by \sqrt[n]{α} ↦ \sqrt[n]{α}^{b} for some b, and the map Gal(M/K) ⊂ ℤ/nℤ is σ ↦ b.

The above observation is Kummer theory: There is a conceptually simple description of the exponent n abelian extensions of K, assuming that all nth roots of unity are in K. Of course, understanding K^{∗}/(K^{∗})^{n} well involves understanding the failure of unique factorization into primes, hence understanding the unit group and class group of the ring of integers of K well.

When the nth roots of unity are not in K, the situation is much more complicated, and is answered by Class Field Theory.

Remark 18.2. A concise general reference about Kummer theory of fields is Birch’s article Cyclotomic Fields and Kummer Extensions in Cassels-Frohlich. For a Galois-cohomological approach to Class Field Theory, see the whole Cassels-Frohlich book.

18.2 Kummer Theory for an Elliptic Curve

Let n be a positive integer, and let E be an elliptic curve over a field K of characteristic coprime to n, and let L = K^{sep}. We mimic the previous section, but for the G_{K}-module E(L) instead of L^{∗}. Consider the exact sequence

0 → E[n] → E ⟶^{[n]} E → 0.

Taking cohomology we obtain an exact sequence

0 → E(K)/nE(K) → H^{1}(K, E[n]) → H^{1}(K, E)[n] → 0.

Unlike the above situation where H^{1}(K, L^{∗}) = 0, the group H^{1}(K, E)[n] is often very large, e.g., when K is a number field, this group is always infinite.

In Kummer theory, we obtained a nice result under the hypothesis that μ_{n} ⊂ K. The analogous hypothesis in the context of elliptic curves is that every element of E[n] is defined over K, in which case

H^{1}(K, E[n]) ≈ Hom(G_{K}, (ℤ/nℤ)^{2}),

where we have used that E[n](L) ≈ (ℤ/nℤ)^{2}, which is a standard fact about elliptic curves, and as usual all homomorphisms are continuous. Another consequence of our hypothesis that E[n](K) = E[n] is that μ_{n} ⊂ K; this later fact can be proved using the Weil pairing, which is a nondegenerate G_{K}-invariant map

E[n] ⊗ E[n] → μ_{n}.

As above, we can interpret the elements φ ∈ Hom(G_{K}, (ℤ/nℤ)^{2}) (modulo an equivalence relation) as corresponding to abelian extensions M of K such that Gal(M/K) ⊂ (ℤ/nℤ)^{2}. Moreover, we have upon fixing a choice of basis for E[n], an exact sequence

0 → E(K)/nE(K) → Hom(G_{K}, (ℤ/nℤ)^{2}) → H^{1}(K, E)[n] → 0,

or, using Kummer theory from the previous section,

0 → E(K)/nE(K) → (K^{∗}/(K^{∗})^{n})^{2} → H^{1}(K, E)[n] → 0.

Another standard fact about elliptic curves—the (weak) Mordell-Weil theorem— is that when K is a number field, then E(K)/nE(K) is finite. Thus when E[n](K) = E[n] , we have a fairly explicit description of H^{1}(K, E)[n] in terms of K^{∗} and E(K). This idea is one of the foundations for using descent to compute Mordell-Weil groups of elliptic curves.

If we restrict to classes whose restriction everywhere locally is 0 we obtain the sequence

0 → E(K)/nE(K) → Sel^{(n)}(E/K) → � (E/K)[n] → 0.

Here

Sel^{(n)}(E/K) = ker (H^{1}(K, E[n]) → ⨁_{all v} H^{1}(K_{v}, E)) ,

and

� (E/K) = ker (H^{1}(K, E) → ⨁_{all v} H^{1}(K_{v}, E)) .

When K is a number field, it is possible to describe Sel^{(n)}(E/K) so explicitly as a subgroup of (K^{∗}/(K^{∗})^{n})^{2} that one can prove that Sel^{(n)}(E/K) is computable.

Theorem 18.3. Given any elliptic curve E over any number field K, and any integer n, the group Sel^{(n)}(E/K) defined above is computable.

It is a major open problem to show that E(K) is computable. A positive solution would follow from the following conjecture:

Conjecture 18.4 (Shafarevich-Tate). The group � (E/K) is finite.

Conjecture 18.4 is extremely deep; for example, it is a very deep (hundreds of pages!) theorem when E/ℚ has “analytic rank” 0 or 1, and is not known for even a single elliptic curve defined over ℚ with analytic rank ≥ 2.

Example 18.5. Consider an elliptic curve E over ℚ of the form y^{2} = x(x − a)(x + b), so that all the 2-torsion of E is ℚ-rational. As above, we obtain an exact sequence

0 → E(ℚ)/2E(ℚ) → ((ℚ^{∗})/(ℚ^{∗})^{2})^{2} → H^{1}(ℚ, E)[2] → 0.

From this diagram and the fact that E(ℚ)/2E(ℚ) is finite, we see that H^{1}(ℚ, E)[2] is infinite. Moreover, given any pair (α, β) of nonzero rational numbers, we can write down an explicit Galois cohomology class in H^{1}(ℚ, E)[2], and given any rational point P ∈ E(ℚ) we obtain a pair of rationals in ((ℚ^{∗})/(ℚ^{∗})^{2})^{2}.

19 Brauer Groups

This lecture is about Brauer groups.

Reference: Chapter X of Serre’s Local Fields.

19.1 The Definition

Let k be a field, and fix a separable closure k^{sep} of k.

Definition 19.1. The Brauer group of k is

Br_{k} = H^{2}(k, (k^{sep})^{∗}).

The Brauer group of a field is a measure of the complexity of the field. It also plays a central role in duality theorems, and in class field theory.

19.2 Some Motivating Examples

(i) Let E be an elliptic curve over k and n a positive integer coprime to char(k). Consider the Weil pairing

E[n] ⊗ E[n] → μ_{n}.

Cup product defines a map

H^{1}(k, E[n]) ⊗ H^{1}(k, E[n]) → H^{2}(k, μ_{n}).

The inclusion μ_{n} ↪ (k^{sep})^{∗} defines a homomorphism

H^{2}(k, μ_{n}) → H^{2}(k, (k^{sep})^{∗}) = Br_{k} .

We thus have a pairing on H^{1}(k, E[n]) with values Br_{k}. It would thus be very handy to understand Brauer groups better.

(ii) If A is a simple abelian variety over k, then R = End(A) ⊗ k is a division algebra over k. Its center is an extension F of k, and R is a central simple F -algebra. As we will see later, the isomorphism classes of central simple F -algebras are in natural bijection with the elements of Br_{F}. It would thus be very handy, indeed, to understand Brauer groups better.

19.3 Examples

Recall that if G is a finite cyclic group and A is a G-module, then Ĥ^{2q}(G, A) ≈ Ĥ^{0}(G, A) and Ĥ^{2q+1}(G, A) ≈ Ĥ^{1}(G, A), a fact we proved by explicitly writing down the following very simple complete resolution of G:

⋯ → ℤ[G] ⟶^{s−1} ℤ[G] ⟶^{N} ℤ[G] ⟶^{s−1} ℤ[G] ⟶^{N} ℤ[G] → ⋯,

where N = ∑ s^{i} is the norm.

Proposition 19.2. The Brauer group of the field ℝ of real numbers has order 2.

Proof. We have ℂ = ℝ^{sep}, and G = Gal(ℂ/ℝ) is cyclic of order 2. Thus

Br_{ℝ} = H^{2}(G, ℂ^{∗}) ≅ Ĥ^{0}(G, ℂ^{∗}) ≈ (ℂ^{∗})^{G}/Nℂ^{∗} ≅ ℝ^{∗}/ℝ_{+}^{∗} ≅ {±1}.

Lemma 19.3. Suppose G is a finite cyclic group and A is a finite G-module. Then

#Ĥ^{q}(G, A) = #Ĥ^{0}(G, A)

for all q ∈ ℤ, i.e., #Ĥ^{q}(G, A) is independent of q.

Proof. Since, as was mentioned above, Ĥ^{2q}(G, A) ≈ Ĥ^{0}(G, A) and Ĥ^{2q+1}(G, A) ≈ Ĥ^{1}(G, A), it suffices to show that #Ĥ^{−1}(G, A) = #Ĥ^{0}(G, A). Let s be a generator of G. We have an exact sequence

0 → A^{G} → A ⟶^{s−1} A → A/(s − 1)A → 0.

Since every term in the sequence is finite,

#A^{G} = #(A/(s − 1)A).

Letting N_{G} = ∑ s^{i} be the norm, we have by definition an exact sequence

0 → Ĥ^{−1}(G, A) → A/(s − 1)A ⟶^{N_{G}} A^{G} → Ĥ^{0}(G, A) → 0.

The middle two terms in the above sequence have the same cardinality, so the outer two terms do as well, which proves the lemma.

Proposition 19.4. If k is a finite field, then Br_{k} = 0.

Proof. By definition,

Br_{k} = H^{2}(k, k̄^{∗}) = \underrightarrow{lim}_{F} H^{2}(F/k, F^{∗}),

where F runs over finite extensions of k. Because G = Gal(F/k) is a finite cyclic group, Lemma 19.3 and triviality of the first cohomology of the multiplicative group of a field together imply that

# H^{2}(F/k, F^{∗}) = #Ĥ^{1}(F/k, F^{∗}) = 1.

Example 19.5. The following field all have Br_{k} = 0.

(i) Let k be any algebraically or separably closed field. Then Br_{k} = 0, obviously, since k^{sep} = k.

(ii) Let k be any extension of transcendance degree 1 of an algebraically closed field. Then Br_{k} = 0. (See §X.7 of Serre’s Local Fields for references.)

(iii) Let k be the maximal unramified extension K^{ur} of a local field K with perfect residue field (e.g., the maximal unramified extension of a finite extension of ℚ_{p}). Then Br_{k} = 0. (See §X.7 of Serre’s Local Fields for references.)

(iv) Let k be any algebraic extension k of ℚ that contains all roots of unity (thus k is necessarily an infinite degree extension of ℚ). Then Br_{k} = 0.

The following theorem is one of the main results of local class field theory.

Theorem 19.6. Let k be a local field with perfect residue field (e.g., a finite extension of ℚ_{p}). Then Br_{k} ≅ ℚ/ℤ.

The following theorem is one of the main results of global class field theory.

Theorem 19.7. Let k be a number field, and for any place v of k, let k_{v} be the completion of k at v, so k_{v} is a p-adic local field, ℝ, or ℂ. We have a natural exact sequence

0 → Br_{k} → ⨁_{v} Br_{k_{v}} \underrightarrow{(x_{v})↦∑ x_{v}} ℚ/ℤ → 0,

We obtain the map to ℚ/ℤ by using Theorem 19.6 to view each Br_{k_{v}} as ℚ/ℤ, and we view Br_{ℝ} = \frac{1}{2}ℤ/ℤ.

19.4 Brauer Groups and Central Simple Algebras

Definition 19.8. Let k be a field. Then a central simple k-algebra is a finite dimensional k-algebra A that satisfies any one of the following equivalent conditions:

(i) A has no nontrivial two-sided ideals, and A has center k.

(ii) The algebra Ak̄ = A ⊗_{k} k̄ is isomorphic to a matrix algebra over \overline{k}.

(iii) There is a finite extension F/k such that A_{F} is isomorphic to a matrix algebra over F.

(iv) A is isomorphic to a matrix algebra over a division algebra D with center k.

We say that two central simple k-algebras are equivalent if the corresponding division algebras D in (iv) above are k-isomorphic. Tensor product endows the set of equivalence classes of central simple k-algebras with the structure of abelian group.

Theorem 19.9. The group B_{k} of equivalence classes of central simple kalgebras is isomorphic to the Brauer group Br_{k}.

The proof of Theorem 19.9 is somewhat involved. We will content ourselves with sketching some of the main ideas; in particular, we will explicitly construct the homomorphism B_{k} → Br_{k}, but will not prove that it is an isomorphism (the argument, which uses descent, is given in Serre’s Local Fields).

Fix a finite Galois extension F of k and let B(n, F/k) be the set of equivalence classes of central simple k-algebras A such that A_{F} ≈ M_{n}(F), where M_{n}(F) is the algebra of n × n matrices over F. Then B is the union of all B(n, F/k) over all n and F.

Given A ∈ B(n, F/k), let φ : A_{F} → M_{n}(F) be a fixed choice of isomorphism. Define a set-theoretic map

f : Gal(F/k) → Aut_{F} (M_{n}(F)) ≈ PGL_{n}(F)

by

f(s) = φ^{−1} ∘ s(φ) = φ^{−1} ∘ s ∘ φ ∘ s^{−1}

Then

[f] ∈ H^{1}(F/k, PGL_{n}(F)),

where this H^{1} is a cohomology set (!).

Proposition 19.10. The above construction A ↦ [f] defines a bijection between B(n, F/K) and H^{1}(F/k, PGL_{n}(F)).

(The above proposition is proved in Serre’s Local Fields.)

Consider the exact sequence

1 → F^{∗} → GL_{n}(F) → PGL_{n}(F) → 1.

There is a well-defined connecting homorphism

H^{1}(F/k, PGL_{n}(F)) → H^{2}(F/k, F^{∗}).

Since H^{2}(F/k, F^{∗}) ⟶^{inf} Br_{k}, we thus obtain a natural map

B(n, F/K) → Br_{k} .

This induces the claimed isomorphism B → Br_{k}.

20 Galois Cohomology of Abelian Varieties

20.1 Principal Homogenous Spaces for Abelian Varieties

See also Pete Clark’s http://math.uga.edu/~pete/wcnotes.pdf.

An abelian variety A over a field k is a projective group variety, i.e., a projective variety that is equipped with a group structure A × A → A and 1_{A} : k → A. Perhaps the first basic theorem about abelian varieties is that their group structure is commutative. We will not prove this here, since it requires too much algebraic geometry (for a complete proof readable by anybody who has read Hartshorne’s Algebraic Geometry, see Milne’s Abelian Varieties article in Cornell-Silverman).

A principal homogenous space for an abelian variety A over a field k is a variety X over k and a morphism ι : A × X → X that satisfies the axioms of a simply transitive group action.

If F is any field such that X(F) ≠ ∅, then A_{F} ≈ X_{F}, so we can view the principal homogenous spaces for A as twists of A as algebraic varieties (not as abelian varieties). Two principal homogenous spaces are equivalent if there is a morphism X → Y such that natural compatibility holds.

Given principal homomogenous spaces X and Y, the Baer sum defines a new principal homogenous space. Define an action of A on X × Y by (a, x × y) = (a, x) × (−a, y). The Baer sum of X and Y is the quotient of X × Y by this action. The diagonal action a.(x × y) = ax × ay then gives the Baer sum the structure of principal homogeneous space for A.

The collection of isomorphism classes of principal homomogenous spaces for a fixed abelian variety A over k equipped with Baer sum is an abelian group, called the Weil-Chatalet group of A, and denoted WC(A/k).

Theorem 20.1 (Lang-Tate, 1958). There is a natural isomorphism WC(A/k) → H^{1}(k, A).

Sketch of Proof. Given a principal homogenous space X for A, we construct an element of H^{1}(k, A) as follows. Since X is a variety of positive dimension, there is a finite extension of k such that X(F) ≠ ∅. Fix a choice of P ∈ X(F). For a ∈ A and x ∈ X, write a + x for the image of (a, x) under the principal homogenous space map A × X → X. Define a map f : G_{k} → A by sending σ ∈ G_{k} to

σ(P) − P

which means “the unique element a ∈ A such that

a + P = σ(P).

The map f is a 1-cocycle because

f(σ) + σf(τ) = σ(P) − P + σ(τ(P) − P) = σ(τ(P)) − P = f(στ),

where we have used the axioms that the principal homogenous space structure satisfy.

Conversely, constructing a principal homogenous space from a cycle f, is called “descent of the base field”. The idea is that we find a finite extension F such that f|_{G_{F}} = 0, i.e., an extension that splits f. Then the data of (A_{F} , f_{G_{F}}) is “descent datum”, which determines an algebraic variety X over k. See Serre Algebraic Groups and Class Fields, Section ???, for more details.

Example 20.2. If A has dimension 1 then A is an elliptic curve. The principal homogenous spaces X for A are genus 1 curves with Jac(X) = A. If A is defined over a number field k, then the nonzero elements of � (A) are in bijection with the set of equivalence classes of principal homogenous spaces X such that X(k_{v}) ≠ ∅ for all places v of k, yet X(k) = ∅. Thus � (A) measures the obstruction to a local-to-global principal.

20.2 Galois Cohomology of Abelian Varieties over Finite Fields

Let A be an abelian variety over a finite field k.

The following theorem was proved by Lang in 1956. A more modern prove is given in the first few sections of Chapter VI of Serre’s Algebraic Groups and Class Fields. Note that Lang actually proved a more general result about algebraic groups.

Theorem 20.3 (Lang, 1956). Let A be any connected algebraic group over a finite field (e.g., an abelian variety). Then H^{1}(k, A) = 0.

Proof. The following proof is based on what Pete Clark posted in the notes mentioned above. This proof has the advantage that it uses techniques that fit very nicely in the context of the rest of this course.

It suffices to show that H^{1}(k, A)[n] = 0 for every positive integer n. The Kummer sequence associated to 0 → A[n] → A → A → 0 is

0 → A(k)/nA(k) → H^{1}(k, A[n]) → H^{1}(k, A)[n] → 0.

It thus sufficies to prove that

#(A(k)/nA(k)) = # H^{1}(k, A[n]).

We have an exact sequence of finite abelian groups

0 → A(k)[n] → A(k) ⟶^{[n]} A(k) → A(k)/nA(k) → 0.

Thus

#A(k)[n] = #(A(k)/nA(k)),

so now we just have to show that

# H^{1}(k, A[n]) = #A(k)[n].

We have

#Ĥ^{0}(F/k, A(F)[n]) = #Ĥ^{1}(F/k, A(F)[n])

for all finite extensions F of k. In particular let F be any extension of k(A[n]) of degree divisible by n. Because the norm map is multiplicative in towers, we have

Tr_{F/k}(A[n]) = Tr_{k(A[n])/k}(Tr_{F/k(A[n])}(A[n])) = Tr_{k(A[n])/k}([n]A[n]) = Tr_{k(A[n])/k}(0) = 0.

Thus

Ĥ^{0}(F/k, A[n](F)) = A(k)[n]/ Tr_{F/k}(A[n]) = A(k)[n],

where here we write Tr instead of the usual “norm” to denote the element ∑ σ^{i}, where Gal(F/k) = ⟨σ⟩. Thus for all finite extensions of M/F, we have

#Ĥ^{1}(M/k, A[n](M)) = #A(k)[n].

By taking compositums, we see that every extension of k is contained in a finite extension of F, so

# H^{1}(k, A[n]) = # \underrightarrow{lim}_{M/F} Ĥ^{1}(M/k, A[n]) = #A(k)[n].

This proves the theorem.

Remark 20.4. When A is an elliptic curve the Hasse bound and Theorem 20.1 imply the theorem. Indeed, any X ∈ WC(A/k) is a genus 1 curve over the finite field k, hence

|#X − #k − 1| ≤ 2\sqrt{#k}.

It follows that #X ≥ #k + 1 − 2\sqrt{#k} > 0.

We have the following incredibly helpful corollary:

Corollary 20.5. If 0 → A → B → C → 0 is an exact sequence of abelian varieties over a finite field k, then 0 → A(k) → B(k) → C(k) → 0 is also exact.

Proof. The cokernel of B(k) → C(k) is contained in H^{1}(k, A) = 0.

Example 20.6. Suppose E is an optimal elliptic curve quotient of J = J_{0}(N) and p ∤ N is a prime. Then for any integer n ≥ 1, the induced natural map

J(𝔽_{p^{n}}) → E(𝔽_{p^{n}})

is surjective. If E[ℓ] is irreducible, one can use Ihara’s theorem to also prove that J(𝔽_{p^{2}})^{ss}(ℓ) → E(𝔽_{p^{2}})(ℓ) is surjective, where J(𝔽_{p^{2}})^{ss} is the group generated by supersingular points.

Corollary 20.7. We have H^{q}(k, A) = 0 for all q ≥ 1. (In fact, we have Ĥ^{q}(k, A) = 0 for all q ∈ ℤ.)

Proof. Suppose F is any finite extension of the finite field k. Then Gal(F/k) is cyclic, so by a result we proved before (lecture 13), we have

#Ĥ^{q}(F/k, A(F)) = #Ĥ^{1}(F/k, A(F)) = 1

for all q ∈ ℤ.

Corollary 20.8. If F/k is a finite extension of finite fields, and A is an abelian variety, then the natural trace map

Tr_{F/k} : A(F) → A(k)

is surjective.

Proof. By Corollary 20.7 and the definition, we have

0 = Ĥ^{0}(F/k, A(F)) = A(k)/ Tr_{F/k}(A(F)).

Let A be an abelian variety over a number field K, and v a prime of K, with residue class field k = k_{v}. The Néron model A of A is a smooth commutative group scheme over the ring O_{K} of integer of K with generic fiber A such that for all smooth commutative group schemes S the natural map

A(S) → A(S_{K})

is an isomorphism. Reducing modulo v we have an exact sequence

0 → A_{k}^{0} → A_{k} → Φ_{A,v} → 0,  (20.1)

where A_{k}^{0} is the connected component that contains the identity and Φ_{A,v} is a finite flat group scheme over k, called the component group of A at v.

Proposition 20.9. For every integer q, we have

Ĥ^{q}(k, A_{k}) = Ĥ^{q}(k, Φ_{A,v}).

Proof. Take Galois cohomology associated to the exact sequence (20.1), and use Corollary 20.7.

21 Duality

WARNING: For the rest of this book, we’re going to let k̄ denote a separable closure of k, since it’s much easier notation to work with (I’ll go back and change the notation above later).

Let k be a field and k̄ a choice of separable closure of k.

21.1 Duality over a Local Field

Let M be any G_{k} = Gal(k̄/k)-module and set

M̂ = Hom(M, k̄^{∗}),

which we give the structure of (left) G_{k}-module by

(g.φ)(a) = g(φ(g^{−1}a)).

To see that this gives M̂ a G_{k}-module structure, note that if g, h ∈ G_{K}, then

((gh).φ)(a) = (gh)(φ((gh)^{−1}a)) = (gh)(φ(h^{−1}g^{−1}a))

and

(g.(h.φ))(a) = g((h.φ)(g^{−1}a)) = g(h(φ(h^{−1}g^{−1}a))).

Theorem 21.1 (Tate Local Duality). Let k be a local field and M a finite G_{k}-module of order coprime to the characteristic of k. Then for r = 0, 1, 2, the cup product pairing

H^{r}(k, M) × H^{2−r}(k, M̂) → H^{2}(k, k̄^{∗}) ≅ ℚ/ℤ

is nondegenerate. Also H^{q}(k, M) = 0 for q ≥ 3.

The proof of this theorem is beyond the scope of this course, since it requires developing too much general machinery. See [Ser97] for complete details.

21.1.1 Example: n-torsion on an elliptic curve

Let k be a local field and M = E[n] the points of order n on an elliptic curve over k. The Weil pairing is a nondegenerate perfect pairing

E[n] ⊗ E[n] → μ_{n},

hence defines a G_{k}-isomorphism E[n] ≅ Ê[n]. Thus for r = 0, 1, 2, we obtain nondegenerate pairings

H^{r}(k, E[n]) × H^{2−r}(k, E[n]) → ℚ/ℤ.

For r = 0, the nondegenerate pairing is

E(k)[n] × H^{2}(k, E[n]) → ℚ/ℤ,

which thus allows us to compute

H^{2}(k, E[n]) ≅ Hom(E(k)[n], ℚ/ℤ) ≈ E(k)[n].

For r = 1, the pairing is

H^{1}(k, E[n]) × H^{1}(k, E[n]) → ℚ/ℤ.  (21.1)

Thus given any element x ∈ H^{1}(k, E[n]) we obtain a homomorphism

φ_{x} : H^{1}(k, E[n]) → ℚ/ℤ,

and φ_{x} = 0 if and only if x = 0.

Tate also proved the following related duality theorem, which gives a very nice way to think about the Galois cohomology of an abelian variety over a local field. Let A be an abelian variety over a local field k, and let A^{∨} denote tha dual abelian variety. When A has dimension 1 or when A is the Jacobian of a curve, then A^{∨} ≅ A.

Theorem 21.2 (Tate). We have H^{q}(k, A) = 0 for q ≥ 2. Moreover, there is a canonical pairing

H^{0}(K, A^{∨}) × H^{1}(K, A) → ℚ/ℤ,

which induces an isomorphism (of discrete groups)

H^{1}(K, A) ⟶^{≅} Hom(A^{∨}(K), ℚ/ℤ).

(All homs are continuous.)

The following proposition is proved using formal groups; it’s Proposition VII.6.3 in [Sil92].

Proposition 21.3. Suppose E is an elliptic curve over a finite extension k of ℚ_{p} and let R be the ring of integers of k. Then E(k) contains a subgroup of finite index that is isomorphic to (R, +).

For simplicity, assume that R = ℤ_{p}. Then we have an exact sequence

0 → ℤ_{p} → E(ℚ_{p}) → M → 0,

where M is a finite group. Applying the Hom(−, ℚ/ℤ) (continuous homomorphisms) Pontryagin duality gives an exact sequence

0 → Hom(M, ℚ/ℤ) → Hom(E(ℚ_{p}), ℚ/ℤ) → Hom(ℤ_{p}, ℚ/ℤ) → 0.

We have

Hom(ℤ_{p}, ℚ/ℤ) ≅ ℚ_{p}/ℤ_{p},

where the isomorphism sends φ : ℤ_{p} → ℚ/ℤ to φ(1) ∈ ℚ_{p}/ℤ_{p} ⊂ ℚ/ℤ. Thus we have an exact sequence

0 → F → H^{1}(ℚ_{p}, E) → ℚ_{p}/ℤ_{p} → 0,

where F is a finite group. So that’s what cohomology of an elliptic curve over a local field looks like:

The group H^{1}(ℚ_{p}, E) has a lot of elements of order a power of p, and not much else.

21.2 Duality over a Finite Field

Let k be a finite field. Then the dualizing module is ℚ/ℤ with trivial action (not k̄^{∗} as above), and we define

M^{∗} = Hom(M, ℚ/ℤ).

Theorem 21.4. For every finite module M and r = 0, 1, cup product induces a nondegenerate pairing

H^{r}(k, M) × H^{1−r}(k, M^{∗}) → ℚ/ℤ.

22 A Little Background Motivation for Étale Cohomology

22.1 Schemes

Schemes are algebraic-geometric objects that generalize the notion of variety and, like manifolds, are built by glueing together special types of objects called affine schemes. The category of commutative rings is equivalent (via a contravariant functor) to a subcategory of the category of schemes, so schemes generalize the notion of commutative rings.

Affine schemes X = Spec(R) are geometric objects attached to commutative rings. Here X is a locally ringed topological space. The set of points of X is the set of prime ideals of the ring R. The closed sets for the (Zariski) topology on X are the subsets

V (I) = {𝔭 ∈ X : I ⊂ 𝔭}

attached to ideals I ⊂ R. To say that X is locally ringed means that there is a sheaf O_{X} of rings on X, i.e., a certain type of functor from the category of open subsets of X to the category of rings. In particular, if U = X −V (I) is an open set with I prime, then

O_{X}(U) = R_{I}

is the localization of R at I, which is obtained by adjoining to R the inverses of the multiplicatively closed subset all elements of R not in I.

Example 22.1. Let X = Spec(K), where K is a field. Then there is exactly one prime ideal, the 0 ideal, so #X = 1, i.e., X is a point. The structure sheaf is O_{X}(X) = K.

Example 22.2. Let X = Spec(ℤ_{p}), where ℤ_{p} is the ring of p-adic integers. Then as a set X = {(0), (p)}, since ℤ_{p} has exactly two prime ideals. There are two nonempty open sets: X itself and U = {(0)} = X − V ((p)). We have O_{X}(X) = ℤ_{p} and O_{X}(U) = R_{(p)} = ℤ_{p}, since everything not in (p) is already invertible in ℤ_{p}. One interesting thing about Spec(ℤ_{p}) is that the natural quotient homomorphism ℤ_{p} → 𝔽_{p} induces a morphism of schemes:

Spec(𝔽_{p}) ↪ Spec(ℤ_{p}).

Example 22.3. The multiplicative group 𝔾_{m} is the scheme

𝔾_{m} = Spec(ℤ[X, Y]/(XY − 1)).

It has a group scheme structure, in that there is a multiplication map 𝔾_{m} × 𝔾_{m} → 𝔾_{m} and an identity section Spec(ℤ) → 𝔾_{m}.

22.2 Étale Cohomology

Inspired by Serre’s introduction of homological techniques into Algebraic Geometry, Grothendieck introduced the idea of étale cohomology in 1958. He was partly motivated by a desire to more deeply understand and prove Weil’s conjectures, which imply the function field analogue of the Riemann Hypothesis (Deligne did indeed prove these conjectures in the 1970s using étale cohomology). Étale cohomology gives a precise meaning to

H_{ét}^{q}(X, F),

where X is a scheme and F is a “sheaf for the étale site on X”, which is a generalization of the notion of module (and also related to the notion of sheaf for the Zariski topology on X). An example of a scheme is X = Spec(K), where K is a field, in which case étale sheaves are in bijection with G_{K}- modules, and

H_{ét}^{q}(X, F) = H_{ét}^{q}(Spec(K), F) = H^{q}(K, F(K̄))

is just the usual concrete notion of Galois chomology that we now know and love.

Thus étale cohomology is an extremely powerful and natural generalization of Galois cohomology.

Remark 22.4. If you know sheaf cohomology in the context of algebraic geometry and the Zariski topology on a scheme (e.g., chapter 3 of Hartshorne’s book Algebraic Geometry), then you’ll find the following interesting:

“The reason that the Zariski topology does not work well is that it is too coarse: it has too few open sets. There seems to be no good way to fix this by using a finer topology on a general algebraic variety. Grothendieck’s key insight was to realize that there is no reason why the more general open sets should be subsets of the algebraic variety: the definition of a sheaf works perfectly well for any category, not just the category of open subsets of a space. He defined étale cohomology by replacing the category of open subsets of a space by the category of étale mappings to a space: roughly speaking, these can be thought of as open subsets of finite unbranched covers of the space. These turn out (after a lot of work) to give just enough extra open sets that one can get reasonable cohomology groups...”

– the Wikipedia article on étale cohomology

23 Étale Cohomology over a DVR

As mentioned above, étale cohomology of Spec(K), when K is a field, is the same thing as Galois cohomology over K. Perhaps the simplest example in which étale cohomology is not just usual Galois cohomology is the case when X = Spec(R) is the spectrum of a discrete valuation ring. In this section, we study this case in explicit detail.

23.1 Discrete Valuation Rings

A local ring is a ring with a unique maximal ideal, and a discrete valuation ring (DVR) is a local principal ideal domain. We let R be a discrete valuation ring (DVR) with field of fractions K and perfect residue field k. Any DVR is equipped with a valuation v : K → ℚ such that R = {α ∈ K : v(α) ≥ 0}.

Example 23.1. We give some examples and non-examples of DVR’s:

(i) For example, we could take R = ℤ_{p} with p-adic valuation, K = ℚ_{p}, and k = 𝔽_{p}. This choice of R is a (topologically) complete local ring.

(ii) We could take

R = ℤ_{(p)} = \{\frac{a}{b} ∈ ℚ : p ∤ b\} ⊂ ℚ

with its p-adic valuation, K = ℚ and k = 𝔽_{p}. Note that this R is not complete, though R is a local ring.

(iii) We could take R = 𝔽_{p}[[t]], the ring of power series in t, and K = 𝔽_{p}((t)) and k = 𝔽_{p}.

(iv) The ring ℤ with a p-adic valuation is not a DVR, since it has more than one maximal ideal.

(v) The ring R = 𝔽_{p}(t)[[X]] does not have perfect residue field, so isn’t allowed as one of the R’s.

23.2 Galois Groups associated to DVR’s

Let K̄ be a separable closure of K and extend the valuation v of R to K̄. Let the subscript of v denote completion with respect to v. Let K^{ur} be the maximal unramified extension of K contained in K̄. The decomposition group is

G_{v} = Gal(K̄_{v}/K_{v}) ⊂ Gal(K̄/K),

the inertia group is

I_{v} = Gal(K̄_{v}/K_{v}^{ur}) ⊂ G_{v},

and we set

G_{k} = Gal(k̄/k).

A standard fact, which one proves in a first course in algebraic number theory, is that we have a natural exact sequence

1 → I_{v} → G_{v} → G_{k} → 1.

(The nontrivial part is that G_{v} → G_{k} is surjective.)

23.3 Galois Modules over a DVR

For any G_{K}-module M, let M^{0} = M^{I_{v}}, which is a G_{k} = G_{v}/I_{v}-module.

Following [Maz73] (Mazur, Notes on Étale Cohomology of Number Fields, 1973, extracted from a course he once gave at Orsay in France), we define the category of Galois modules over R as follows. The objects are diagrams

N ⟶^{φ} M,

where N is a G_{k}-module, M is a G_{K}-module, and φ is a homomorphism of abelian groups that sends N into M^{0} ⊂ M, and when viewed as a map of N into M^{0} is a homomorphism of G_{k}-modules. A morphism is a commuting square

\xymatrix{N \ar[r]^{φ} \ar[d] & M \ar[d] \\ N′ \ar[r]^{ψ} & M′}

We will also frequently write the above diagram as a tuple

(M, N, φ),

where of course φ : N → M satisfies the above conditions.

In Section 24.3 we will define the category of “abelian sheaves on the étale site over Spec(R)”.

Theorem 23.2. There is a natural equivalence of categories between the category S_{R} of triples (M, N, φ) and the category of “abelian sheaves for the étale topology on Spec(R)”.

Thus studying cohomology in the category S_{R} of Galois modules over R is exactly the same as studying étale cohomology of abelian sheaves on the étale site over Spec(R). However, it is more explicit and concrete.

Example 23.3. Let M be any G_{K}-module and let N = M^{0} and id : N ↪ M be the natural inclusion. Then (M, M^{0}, id) is a Galois module over R.

Example 23.4. Let A be any abelian group equipped with trivial Galois action, and set N = M = A above and φ the identity map. In terms of a triple, we write this element of S_{R} as (A, A, 1_{A}). We call such a Galois module constant. For example, we denote by ℤ the Galois module obtained in this way by taking A = ℤ.

Example 23.5. Let

U = {α ∈ (K^{ur})^{∗} : v(α) = 0}

be the group of units in the ring of integers of the maximal unramified extension K^{ur} of K. Then

𝔾_{m,R} = (K̄^{∗}, U, id).

is a Galois module, since (K̄^{∗})^{0} = (K^{ur})^{∗} contains U.

23.4 The Natural Functors

In order to better understand S_{R}, we introduce several natural functors. We define the following functors: pullbacks:

i^{∗} : (M, N, φ) ↦ N; j^{∗} : (M, N, φ) → M;

push forward:

i_{∗} : N ↦ (0, N, 0); j_{∗} : M ↦ (M, M^{0}, id).

extension by zero over a closed point:

j_{!} : M ↦ (M, 0, 0)

sections with support on a closed point:

i^{!} : (M, N, φ) ↦ ker(φ)

Thus we have defined functors as follows:

i^{∗} : S_{R} → S_{k}, j^{∗} : S_{R} → S_{K}, i_{∗} : S_{k} → S_{R},
j_{∗} : S_{K} → S_{R}, j_{!} : S_{K} → S_{R}, i^{!} : S_{R} → S_{k}.

The functors i^{∗}, i_{∗}, j^{∗}, j_{!} are exact, and j_{∗} and i^{!} are left exact.

23.5 Cohomology of Galois Modules over a DVR

Let ℤ be the Galois module from Definition 23.4 and let F be any Galois module. In this section, we define

H^{q}(R, F)

for Galois modules F over a DVR R. This is our first generalization of Galois cohomology in the direction of étale cohomology.

Definition 23.6 (Cohomology over a DVR). Let

H^{q}(R, F) = Ext_{S_{R}}^{q}(ℤ, F)

Letting

Γ_{R}(F) = Hom_{S_{R}}(ℤ, F),

we have

Ext_{S_{R}}^{q}(ℤ, F) = (R^{q}Γ_{R})(F),

where R^{q} denotes the qth right derived functor of the left exact functor Γ_{R}. Thus to compute H^{q}(R, F) we find a resolution of either ℤ or F in the category S_{R} of Galois modules over R, then apply the appropriate Hom functor in the category S_{R} and take the (co-)homology of the resulting complex.

The following result generalizes Hilbert’s theorem 90, and also emphasizes how étale cohomology and Galois cohomology can differ.

Theorem 23.7. We have

H^{q}(R, 𝔾_{m,R}) = \begin{cases}R^{∗} & if q = 0, \\ 0 & otherwise. \end{cases}

24 The Étale Topology

See §III.3 of Hartshorne for the following general definition of étale morphism f : X → Y of schemes: f is étale if it is smooth of relative dimension 0. Since “smooth” and “relative dimension” take some work to define, we will instead give a more concrete definition. By Corollary I.3.16 in Milne’s published book [Mil80], we may define étale as follows.

Definition 24.1. A morphism f : X → Y of schemes is étale if for every x ∈ X, there exists open affine neighborhoods V = Spec(C) of x and U = Spec(A) of y = f(x), and polynomials P, b ∈ A[t] with P monic, such that

C ≈ A[t, u]/(P, bu − 1),

with P′ a unit in C. (Here the map A → C is induced by f.)

Figure 24.1: Typical Result of Google Image Search for “Étale”

Thus locally on rings, any étale morphism is obtained by adjoining an element and inverting an element to remove any ramified primes.

Example 24.2. Let X = Spec(ℤ[i, 1/2]) and Y = Spec(ℤ). Then the map ℤ ↪ ℤ[i, 1/2] induces an étale morphism X → Y, since taking P = t^{2} + 1 and b = 2t, we have

ℤ[i, 1/2] ≅ ℤ[t, u]/(t^{2} + 1, 2tu − 1),

and P′ = 2t is a unit in ℤ[t, u]/(t^{2} + 1, 2tu − 1). Note that if we took b = 1 instead of b = 2, then P′ would not be a unit. More generally, if P is any squarefree monic polynomial, then ℤ[t, u]/(P, P′ · tu − 1) is étale over ℤ.

An incredibly useful fact about étale morphisms is that they are “closed under base extension”, in the sense of the following theorem:

Theorem 24.3. If f : X → Y is étale and g : Z → Y is any morphism of schemes at all, then the map f′ : Z ×_{Y} X → Z obtained by fiber product is étale.

\xymatrix{Z ×_{Y} X \ar[r] \ar[d]^{f′} & X \ar[d]^{f} \\ Z \ar[r]^{g} & Y}

Proof. Hartshorne.

Remark 24.4. It is frequently the case in algebraic geometry that interesting classes of morphisms are “closed under base extension”.

Example 24.5. Suppose R is a DVR with fraction field K and residue class field k. Suppose L is any finite seperable extension of K. Then the morphism Spec(L) → X = Spec(R) induced by R ↪ K ⊂ L is an étale morphism. To see this, in Definition 24.1, take V = Spec(K), which is an affine open subset of X = Spec(R), as {(0)} is open, and the map Spec(K) ↪ Spec(R) has image {(0)}. Since L is finite separable, we have L = K[t]/(P) for some polynomial P with 0 ≠ P′ ∈ L.

24.1 Special Cases of Interest

Suppose X = Spec(R), where R is a Dedekind domain, DVR, or field. Suppose R ⊂ E is an extension (with E of the same type as R, e.g., Dedekind domain, DVR, or field) with E finitely generated as an R-module. In case R is a Dedekind domain, suppose 0 ≠ b ∈ E and let

R′ = E_{b} = E[1/b]

be the localization of E away from b. Geometrically

Spec(E_{b}) = Spec(E) − { finitely many points },

where the omitted points are the prime ideals that contain b, i.e., those that divide the ideal generated by b. Then the extension R ⊂ R′ induces an étale morphism Spec(R′) → X if and only if every prime of R′ is unramified over R.

If R is a field, then Spec(E) → X is étale if and only if E is a finite separable extension of R (note that E just has to be a product of fields). When R and E are both DVR’s, then Spec(E) → X is étale if E is unramified over R.

24.2 Étale Coverings

For X = Spec(R), let Ét(X) denote the category of all schemes Y/X where the structure morphism Y → X is étale.

An étale covering of an object Y in Ét(X) is a collection {U_{i} ⟶^{f_{i}} Y } of U_{i} ∈ Ét(X) such that the union of the images equals Y, i.e.,

Y = ⋃_{i} f_{i}(U_{i}).

Example 24.6. Let Y = X = Spec(ℤ), let U_{1} = ℤ[i][\frac{1}{2}] and U_{2} = ℤ[ζ_{3}][\frac{1}{3}]. Then U_{1} → Y and U_{2} → Y are étale morphisms and {U_{1}, U_{2}} is an étale covering of Y.

24.3 Étale Sheaves

Definition 24.7 (Étale Site on X). The (small) étale site on X is the category Ét(X) with “topology” given by étale coverings.

Definition 24.8 (Étale Presheaf). An étale presheaf on X is a contravariant functor P from Ét(X) to the category of abelian groups.

Thus P associates to each étale morphism f : Y → X an abelian group P (Y). Moreover, if Y → Y′ is a morphism of objects in Ét(X), there is a corresponding morphism of abelian groups res_{Y,Y′} : P (Y′) → P (Y), which we think of as “restriction maps”.

Example 24.9. Suppose G is a commutative group scheme over a scheme X. Define a functor on Ét(X) by sending a scheme Y/X to the abelian group

G(Y) = Hom_{Ét(X)}(Y, G) (scheme morphisms over X).

Then this functor is an étale presheaf. As a concrete example, we can take G to be the multiplicative group scheme 𝔾_{m}, the additive group scheme 𝔾_{a}, or the Néron model A over X = Spec(R) of an abelian variety A over Spec(K).

Definition 24.10 (Étale Sheaf). An étale sheaf on X is a presheaf P such that for all covers {U_{i} → U} of all objects U ∈ Ét(X), the sequence

P (U) → ∏_{i} P (U_{i}) → ∏_{i,j} P (U_{i} ×_{U} U_{j})

is exact. Here, the first morphism sends a section s ∈ P (U) to the sequence (res_{U_{i},U}(s))_{i}. The second map sends s_{i} ∈ P (U_{i}) to the tuple of differences

res_{U_{i}×_{U} U_{j},U_{i}}(s) − res_{U_{i}×_{U} U_{j},U_{j}}(s) ∈ P (U_{i} ×_{U} U_{j})

Thus P is a sheaf if it is completely determined by its restriction to a covering, and any compatible family of sections on a covering arises from a global section.

Let S_{X} denote the category of all étale sheaves on X.

Theorem 24.11. If G is a commutative group scheme over a scheme X (see Example 24.9 above) then the presheaf defined by G is in fact a sheaf for the étale site on X.

Proof. See [Mil80, Cor. 1.7, page 52].

24.4 Direct and Inverse Image Functors

Suppose π : X′ → X is a morphism of schemes. Then π induces a functor

Ét(X) → Ét(X′),

that sends an étale morphism Y → X to its base change Y ×_{X} X′ → X′:

\xymatrix{Y ×_{X} X′ \ar[r] \ar[d] & Y \ar[d] \\ X′ \ar[r] & X}

Next we define the direct image functor π_{p} on presheaves. Given an étale presheaf P′ on X′ we obtain an étale presheaf π_{p}(P′) on X, which for U ∈ Ét(X), is given by

(π_{p}(P′))(U) = P′(U_{X′}).

Here U_{X′} = U ×_{X} X′ is the fiber product, which is an étale cover of X′:

\xymatrix{U_{X′} \ar[r] \ar[d] & U \ar[d] \\ X′ \ar[r] & X}

We continue to fix an étale morphism f : X′ → X. Given an étale presheaf P on X, the inverse image functor π^{p} associates to P an étale presheaf P′ = π^{p}(P) on X′. By abstract nonsense, the functor π^{p} is completely determined by the assertion that it is the adjoint of π_{p}, i.e.,

Hom(π^{p}(P), Q) ≈ Hom(P, π_{p}(Q)),

where the homsets are in the categories of étale presheaves for X′ and X, respectively. Explicitly, for U′ ∈ Ét(X′)

(π^{p}(P))(U′) = \underrightarrow{lim}_{U} P (U),

where the limit is over all commuting squares

\xymatrix{U′ \ar[r] \ar[d] & U \ar[d] \\ X′ \ar[r] & X}

with U → X in Ét(X). Note that these are merely commuting squares; we do not require that U′ = X′ ×_{X} U. (See [Mil80, pg. 57] for more details about the precise meaning of this direct limit, e.g., what the maps are.)

Proposition 24.12. Both π_{p} and π^{p} are exact functors.

Proof. See [Mil80, II.2.6, pg. 59].

24.5 Stalks

Let X be a scheme and x ∈ X a point of the underlying topological space of X, so there is an affine open subset U = Spec(A) ⊂ X such that x corresponds to a prime ideal 𝔭 ⊂ A. The field associated to x is k(x) = Frac(A/𝔭), and we let x̄ = Spec(k̄), where k̄ is a fixed choice of separable closure of k(x). The inclusion k(x) ↪ k̄ induces a scheme map

x̄ ⟶^{u_{x}} X.

The functor F ↦ F (x̄) is an equivalence of categories between the category of étale sheaves on x̄ and the category of abelian groups.

Let P be an étale presheaf on X. The stalk of P at x̄ is

Px̄ = (u_{x}^{p}P)(x̄)

From the definition of u_{x}^{p}, we see that

(u_{x}^{p}P)(x̄) = \underrightarrow{lim} P (U)

where the limit is over all commutative triangles

\xymatrix{U \ar[d] & x̄ \ar[l] \ar[dl]^{u_{x}} \\ X}

with U ∈ Ét(X). (Think of these as the “open sets U that contain x”.)

The following proposition asserts that any question about an étale sheaf can be attacked by instead studying the stalks of the sheaf. This is why sheaves are much better to work with than presheaves.

Proposition 24.13. Let F be an étale sheaf on X. Then for U ∈ Ét(X), a section s ∈ F (U) is completely determined by the stalks sx̄ of s at all points x̄.

Proof. Suppose s, s′ are two sections and sx̄ = (s′)x̄ for all x̄. Then (s−s′)x̄ = 0 for all x̄, so [Mil80, Prop. II.2.10, pg. 60] implies that s−s′ = 0. The proof of Proposition II.2.10 in Milne is just an easy application of the definitions and sheaf property.

The Galois group G_{k(x)} = Gal(k̄/k(x)) acts on the abelian group Px̄, so the stalk Px̄ is an element of a Galois module. In light of Proposition 24.13, sections of étale sheaves are thus “compatible” families of elements of all of the natural Galois modules attached to the points x ∈ X.

Example 24.14. Suppose X = Spec(R) with R a DVR with fraction field K and perfect residue field k. Then as a set X = {𝔭, (0)} has two points x_{p} = 𝔭 and x_{0} = (0). If P is an étale sheaf on X, then the stalks of P are M = Px̄_{0} and N = Px̄_{p}, which are G_{K} and G_{k}-modules, respectively. This defines the M and N in the equivalence of categories between the category S_{R} of triples (M, N, φ) and the category of étale sheaves on X. The map φ then encodes the compatibility alluded to above; see Section 26.2 for precisely how φ is defined.

Using stalks one proves the following theorem (see [Mil80, Thm. II.2.11, pg. 61]) via a construction involving direct and inverse image functors to reduce to the case of stalks:

Theorem 24.15. For any étale presheaf P on X, there is an étale sheaf a(P) on X (called the étale sheafification of P) and a morphism φ : P → a(P) such that for any sheaf F and morphism φ′ : P → F there is a unique map ψ : a(P) → F making the following diagram commute:

\xymatrix{P \ar[rr]^{φ} \ar[dr]^{φ′} &  & a(P) \ar[dl]_{ψ} \\  & F}

24.6 Pullback and Pushforward of Étale Sheaves

Suppose π : X′ → X is a morphism of schemes. If F′ is an étale sheaf on X′, then the pushforward of F′ is the sheaf

π_{∗}(F′) = π_{p}(F′),

which is an étale sheaf on X. If F is an étale sheaf on X, then the pullback is the sheaf

π^{∗}(F) = a(π^{p}(F)),

which is an étale sheaf on X′. These are adjoint functors, which means that in the appropriate categories of étale sheaves we have

Hom(F, π_{∗}(F′)) ≅ Hom(π^{∗}(F), F′).

The functor π^{∗} is exact and π_{∗} is left exact. In general, π_{∗} is not right exact (even though π_{p} is).

25 Étale Cohomology

Before definining étale cohomology, we recall an import theorem. Recall that an object I in an abelian category C is injective if the contravariant functor Hom_{C}(−, I) is exact. Suppose that C has enough injectives, i.e., for every object X in C there is a monomorphims X → I for some injective I.

Theorem 25.1. If f : C → D is any left exact functor, then there is an essentially unique collection of right derived functors R^{i}f : C → D such that

(i) R^{0}f = f,

(ii) (R^{i}f)I = 0 if I is injective and i ≥ 1,

(iii) Long exact sequences: if 0 → A → B → C → 0 is exact, then there is a long exact sequence

⋯ → (R^{i}f)B → (R^{i}f)C ⟶^{δ} (R^{i+1}f)A → (R^{i+1}f)B → · · · .

Theorem 25.2. Let X be any scheme. The category of étale sheaves on X has enough injectives.

Proof. The following proof is from [Mil80, §III.1]. Let u_{x} : x̄ → X be a geometric point of X. The category Sx̄ of étale sheaves on x̄ is isomorphic to the category Ab of abelian groups, so has enough injectives. Let F ∈ S_{X} be an étale sheaf, and choose for each x ∈ X an embedding u_{x}^{∗}F ↪ F′_{x} of u_{x}^{∗}F into an injective sheaf. Define

F^{∗} = ∏_{x∈X}(u_{x})_{∗}u_{x}^{∗}F

and

F^{∗∗} = ∏_{x∈X}(u_{x})_{∗}F′_{x}.

Then the canonical maps F → F^{∗} and F^{∗} → F^{∗∗} are both monomorphisms, and F^{∗∗} is an injective object because products of injective objects are injective.

Let X be a scheme and consider étale sheaves F on X. The global sections functor S_{X} → Ab

Γ(X, −) : F ↦ F (X)

is left exact.

Definition 25.3 (Étale Cohomology). The Étale cohomology groups H^{i}(X_{ét}, F) are the right derived functors of the global sections functor. Thus

H^{i}(X_{ét}, −) = R^{i}Γ(X, −).

26 Galois Modules and Étale Sheaves

For essentially the rest of this course, we will only consider the the spectrums of DVR’s and fields. For a scheme X, we continue to let S_{X} denote the category of étale sheaves on X. For a ring R we let S_{R} denote the category of Galois modules over R, which is a category of triples (M, N, φ) that we have only defined when R is a field or DVR (with perfect residue field).

26.1 The Spectrum of a Field

Let K be a field and X = Spec(K). Suppose F ∈ S_{X} is an étale sheaf. If M/K is a finite separable extension, then Spec(M) → X is an étale covering, and F (Spec(M)) is defined. The only other étale coverings are of the form (disjoint union)

∐ Spec(M_{i}) → Spec(K),

which by the sheaf condition are completely determined by the groups F (Spec(M_{i})).

Define a functor S_{X} → S_{K} by associating to an (abelian) étale sheaf F ∈ S_{X} the G_{K}-module

\underrightarrow{lim}_{M/K} F (Spec(M)),

where the limit is over all finite field extensions M of K contained in K̄ (the separable closure).

Theorem 26.1. The above functor is an equivalence of categories between S_{Spec(K)} and S_{K}.

Proof. See [Mil80, Thm. 1.9, pg. 53] or [Maz73, Example 1, pg. 532] This can be proved by defining a functor in the other direction S_{K} → S_{Spec(K)}, and being careful about what happens with morphisms.

Remark 26.2. The above functor S_{Spec(K)} → S_{K} is just the functor that associates to a sheaf F its stalk at the point of Spec(K) (see Section 24.5).

26.2 The Spectrum of a DVR

In this section, we revisit and make more explicit Theorem 23.2, which we restate as follows. Recall that S_{R} denotes the category of triples (M, N, φ) where M is a G_{K}-module, N is a G_{k}-module and φ : N → M^{0} = M^{I_{v}} is a G_{k}-module homomorphism (where I_{v} is the inertia group).

Let X = Spec(R) and consider the category S_{X} of étale sheaves on X. Define a functor S_{X} → S_{R} by

F ↦ (M_{F} , N_{F} , φ_{F}),

as follows.

• Define M_{F}: The open immersion

j : Spec(K) ↪ X

induces by pullback a functor j^{∗} : S_{X} → S_{Spec(K)}. We describe this functor explicitly as follows. If F ∈ S_{X} and L/K is a finite extension (with L contained in K̄), then j^{∗}(F)(Spec(L)) = F (Spec(L)), where Spec(L) ∈ Ét(X) via the composition

Spec(L) → Spec(K) ⟶^{j} X.

We define M_{F} to be the G_{K}-module obtained by the composition of functors,

S_{X} ⟶^{j^{∗}} S_{Spec(K)} ⟶^{≅} S_{K},

where j^{∗} is as above and and the functor S_{Spec(K)} → S_{K} is the equivalence of categories from Theorem 26.1. Thus

M_{F} = \underrightarrow{lim}_{L/K as above}F (Spec(L)).

• Define N_{F}: Let

N_{F} = \underrightarrow{lim}_{L/K unramified}F (Spec(O_{L})),

where L runs through all finite extensions of K contained in K^{ur}, and O_{L} is the integral closure of R in L, i.e., the ring of integers of L. We regard N_{F} as a G_{k} ≅ Gal(K^{ur}/K)-module since Gal(K^{ur}/K) acts naturally and compatibly on each group F (Spec(O_{L})), since F is a functor.

• Define φ_{F}: The morphism φ_{F} is induced by the restriction maps on

sheaves

F (Spec(O_{L})) → F (Spec(L)),

which are induced by the inclusion Spec(L) ↪ Spec(O_{L}).

Remark 26.3. The Galois modules M_{F} and N_{F} can alternatively be defined using stalks (see Example 24.14).

Theorem 26.4. The functor F ↦ (M_{F} , N_{F} , φ_{F}) is an equivalence of categories.

Proof. See [Maz73, pg. 533], where Mazur in turn cites “the decomposition lemma”.

27 Étale Cohomology over a DVR

Our immediate goal is to master étale cohomology over a DVR, by understanding various exact sequences, some of which relate étale cohomology to Galois cohomology.

As usual, let R be a DVR with perfect residue field k and field of fractions K, and let X = Spec(R). Let F ∈ S_{R} be a Galois module over R, which we can equivalently view as an étale sheaf in S_{X} by Theorem 26.4. Also, we continue to let

i : Spec(k) ↪ X and j : Spec(K) → X

denote the natural closed embedding and open immersion, respectively, which induce many functors (see Section 23.4): i^{∗}, j^{∗}, i_{∗}, j_{∗}, j_{!}, i^{!}, the functors i^{∗}, i_{∗}, j^{∗}, j_{!} are exact, and j_{∗} and i^{!} are left exact.

Proposition 27.1. We have an exact sequence of étale sheaves

0 → j_{!}j^{∗}F → F → i_{∗}i^{∗}F → 0.  (27.1)

Proof. View F as the triple (M, N, φ). Using the definitions of the functors in the sequence from Section 23.4, we see that the sequence in the statement of the proposition is:

\xymatrix{0 \ar[r] & 0 \ar[r] \ar[d] & N \ar[r] \ar[d]_{φ} & N \ar[r] \ar[d] & 0 \\ 0 \ar[r] & M \ar[r] & M \ar[r] & 0 \ar[r] & 0.}

Since each row is exact, the claim follows.

The long exact sequence associated to (27.1) is:

⋯ → H^{r}(R, j_{!}j^{∗}F) → H^{r}(R, F) → H^{r}(R, i_{∗}i^{∗}F) → ⋯  (27.2)

Thus it would be extremeley helpful if we could better understand both

H^{q}(R, j_{!}j^{∗}F) = H^{q}(R, (M, 0, 0))

and

H^{q}(R, i_{∗}i^{∗}F) = H^{q}(R, (0, N, 0)).

You might hope that H^{q}(R, (M, 0, 0)) is H^{q}(K, M) and H^{q}(R, (0, N, 0)) is H^{q}(k, N), but this is WRONG in general. However, there are relationships, which we can make precise using compact étale cohomology.

Recall that

H^{q}(R, F) = Ext_{S_{R}}^{q}(ℤ, F),

since

Hom_{S_{R}}(ℤ, F) = Hom_{S_{R}}((ℤ, ℤ, id), (M, N, φ)) = N^{G_{k}} = F (R) = Γ(X, F).

(27.3)

We use the above observation about how H^{q} is defined to motivate the introduction of “compact cohomology”, which will help us to better understand the cohomology of j_{!}j^{∗}F and i_{∗}i^{∗}F. Note that if ℤ is the constant sheaf in S_{k}, then

i_{∗}ℤ = (0, ℤ, 0).

Definition 27.2 (Compact Cohomology). Define the compact étale cohomology groups by

H_{comp}^{q}(R, F) = Ext_{S_{R}}^{q}(i_{∗}ℤ, F).

Because there is a long exact sequence of Ext groups, we have a long exact sequence of compact cohomology, etc. (i.e., the H_{comp}^{q}(R, −) are derived functors). However, the best part is that there is a relative cohomology exact sequence.

Proposition 27.3. Let F ∈ S_{R} be any sheaf. Then we have an exact sequence

⋯ → H_{comp}^{q}(R, F) → H^{q}(R, F) → H^{q}(K, j^{∗}F) → ⋯

Proof. In this proof, we write Ext = Ext_{S_{R}}. We also let ℤ denote the constant sheaf in either S_{k} and S_{R}, with the context making which clear. By Proposition 27.1, we have an the exact sequence

0 → j_{!}j^{∗}ℤ → ℤ → i_{∗}i^{∗}ℤ → 0

in the category S_{R}. Apply the Ext^{q}(−, F) functor to this exact sequence, to obtain a long exact sequence

⋯ → Ext^{q}(i_{∗}i^{∗}ℤ, F) → Ext^{q}(ℤ, F) → Ext^{q}(j_{!}j^{∗}ℤ, F) → ⋯

We have i_{∗}i^{∗}ℤ = i_{∗}ℤ, so Ext^{q}(i_{∗}i^{∗}ℤ, F) = H_{comp}^{q}(R, F), and of course by definition Ext^{q}(ℤ, F) = H^{q}(R, F).

It thus remains to show that

Ext^{q}(j_{!}j^{∗}ℤ, F) ≅ H^{q}(K, j^{∗}F).  (27.4)

Since j_{!}j^{∗}ℤ = (ℤ, 0, 0), we have

Hom_{S_{R}}(j_{!}j^{∗}ℤ, F) = Hom_{S_{R}}((ℤ, 0, 0), (M, N, φ))
≅ Hom_{S_{K}}(ℤ, M) = H^{0}(K, M) ≅ H^{0}(K, j^{∗}F).

Also, recall that the functor j^{∗} is exact (see, e.g., [Maz73, pg. 525] or just note that this is easy to see from the definition in terms of triples). The functor

F → H^{q}(K, j^{∗}F)

is thus a delta functor (since j^{∗} is exact and H^{q}(K, −) is a delta functor), and for q = 0 the above functor agrees with Hom_{S_{R}}(j_{!}j^{∗}ℤ, −). By uniqueness of derived functors we have compatible isomorphisms as in (27.4), at least as long as we show that j^{∗} sends injectives to acyclics!, which we do in the next paragraph. In fact, we will show that if I is injective, then j^{∗}I is also injective.

Claim: j^{∗} preserves injectives. Suppose I = (M, N, φ) is an injective element of S_{R}, so Hom_{S_{R}}(−, I) is exact. Suppose 0 → M_{0} → M_{1} → M_{2} → 0 is an exact sequence in S_{K}. Consider the exact sequence

0 → j_{!}(M_{0}) → j_{!}(M_{1}) → j_{!}(M_{2}) → 0.

Then the sequence

0 → Hom_{S_{R}}(j_{!}(M_{2}), I) → Hom_{S_{R}}(j_{!}(M_{1}), I) → Hom_{S_{R}}(j_{!}(M_{0}), I) → 0

is exact (since I is injective). But for a = 0, 1, 2,

Hom_{S_{R}}(j_{!}(M_{a}), I) = Hom((M_{a}, 0, 0), (M, N, φ)) ≅ Hom_{S_{K}}(M_{a}, M) ≅ Hom_{S_{R}}(M_{a}, j^{∗}I).

(In fact, we have just proved that j_{!} and j^{∗} are adjoint functors.) Thus

0 → Hom_{S_{K}}(M_{2}, I) → Hom_{S_{K}}(M_{1}, I) → Hom_{S_{K}}(M_{0}, I) → 0

is exact, as required.

We introduced H_{comp}^{q}(R, F) because it is useful for computing the cohomology of certain sheaves in S_{R}. For example, it nails down shreaks:

Proposition 27.4. For any G_{K} module M and any q ≥ 1, we have

H_{comp}^{q}(R, j_{!}M) ≅ H^{q−1}(K, M).

and

H^{q}(R, j_{!}(M)) = 0.

Proof. If we take F = j_{!}M in Proposition 27.3, then we see that the second assertion that for all q ≥ 0, we have H^{q}(R, j_{!}M) = 0, implies the first assertion about H_{comp}^{q}. First, we start with q = 0. We have j_{!}M = (M, 0, 0), so as in (27.3),

H^{0}(R, j_{!}M) = Γ(Spec(R), (M, 0, 0)) = Hom(ℤ, (M, 0, 0)) = 0.

Recall that the functor j_{!} is exact (as asserted in Section 23.4). As Mazur remarks on [Maz73, pg. 529], “one is tempted to use that j_{!} is exact, and that H^{0}(R, j_{!}) is the zero functor. However, one must check that j_{!}I is acyclic for cohomology over R, whenever I is injective”. Thus, at this point, as in the proof of Proposition 27.3, it suffices to show that if I ∈ S_{K} is injective, then H^{q}(R, j_{!}I) = 0. To do this, consider the exact sequence in S_{R}:

\xymatrix{0 \ar[r] & 0 \ar[r] \ar[d] & I^{0} \ar[r] \ar[d]_{φ} & I^{0} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & I \ar[r] & I \ar[r] & 0 \ar[r] & 0,}

which we may alternatively write as

0 → j_{!}I → j_{∗}I → i_{∗}i^{∗}j_{∗}I → 0,  (27.5)

We first establish that this is an injective resolution of j_{!}I.

Claim: j_{∗} preserves injective. Suppose I ∈ S_{K} is injective and 0 → F_{0} → F_{1} → F_{2} → 0 is an exact sequence in S_{R}. The functor j^{∗} is exact, so 0 → j^{∗}F_{0} → j^{∗}F_{1} → j^{∗}F_{2} → 0 is an exact sequence in S_{K}. By our hypothesis that I ∈ S_{K} is injective, the sequence

0 → Hom(j^{∗}F_{2}, I) → Hom(j^{∗}F_{1}, I) → Hom(j^{∗}F_{0}, I) → 0

is exact. Because j^{∗} and j_{∗} are adjoint, the sequence

0 → Hom(F_{2}, j_{∗}I) → Hom(F_{1}, j_{∗}I) → Hom(F_{0}, j_{∗}I) → 0

is exact. Thus j_{∗}I is injective.

WARNING: In [Maz73, pg. 529] Mazur says in the proof that “the thing to check is that i^{∗} preserves injectives”. (See Remark 27.7 below.) Fortunately, something close enough is true:

Claim: i_{∗}i^{∗}j_{∗}I is injective. The claim is that the functor on S_{R},

F ↦ Hom(F, i_{∗}i^{∗}j_{∗}I)

is exact. Using that i_{∗} and i^{∗} are adjoint, and writing F = (M, N, φ), we have

Hom(F, i_{∗}i^{∗}j_{∗}I) ≅ Hom(i^{∗}F, i^{∗}j_{∗}I) ≅ Hom(N, I^{0}).

The functor i_{∗} is exact, so it suffices to show that the functor

N → Hom(N, I^{0})

on G_{k} = G_{v}/I_{v}-modules is exact, assuming I is an injective G_{K}-module and recalling that I^{0} = I^{I_{v}} is the submodule fixed by the inertia group. Suppose 0 → N_{0} → N_{1} → N_{2} → 0 is an exact sequence of G_{v}/I_{v}-modules. Viewing it as an exact sequence of G_{v}-modules, we see that

0 → Hom(N_{2}, I) → Hom(N_{1}, I) → Hom(N_{0}, I) → 0  (27.6)

is exact, since I is injective. However, since each N_{i} is fixed by I_{v}, the above homomorphisms all have image in I^{0}, i.e., Hom(N_{a}, I) ≅ Hom(N_{a}, I^{0}). Thus the sequence (27.6) with each I replaced by I^{0} is exact in the category of G_{k}-modules, which proves the claim.

Finally, we compute H^{q}(R, j_{!}I) using (27.5). We have

H^{0}(R, j_{∗}I) = Hom(ℤ, (I, I^{0}, id)) = (I^{0})^{G_{k}},

and

H^{0}(R, i_{∗}i^{∗}j_{∗}I) = Hom(ℤ, (0, I^{0}, 0)) = (I^{0})^{G_{k}}.

Since j_{∗}I and i_{∗}i^{∗}j_{∗}I are injective, for q ≥ 1 we have

H^{q}(R, j_{∗}I) = H^{q}(R, i_{∗}i^{∗}j_{∗}I) = 0.

The long exact sequence associated to (27.5), then implies that H^{q}(R, j_{!}I) = 0 for q ≥ 0.

Proposition 27.5. For any G_{k}-module N and all q, we have

H_{comp}^{q}(R, i_{∗}N) ≅ H^{q}(k, N).

Proof. Recall that the functor i_{∗} is exact and is adjoint to the functor i^{∗}, which is exact, so we have

Hom(i^{∗}X, Y) = Hom(X, i_{∗}Y),

hence i_{∗} preserves injectives (this is exactly the same argument as in some of the claims above). Thus

H_{comp}^{q}(R, i_{∗}N) = Ext_{S_{R}}^{q}(i_{∗}ℤ, i_{∗}N) = Ext_{S_{k}}^{q}(ℤ, N) = H^{q}(k, N).

The following theorem “computes” étale cohomology over R in terms of Galois cohomology over k. Think of this theorem as saying “étale cohomology over a DVR is just unramified Galois cohomology.”

Theorem 27.6. Let F = (M, N, φ) ∈ S_{R}. Then for all q ≥ 0, we have a functorial isomorphism

H^{q}(R, F) ≅ H^{q}(k, i^{∗}F) = H^{q}(k, N).

This isomorphism is induced by the natural map

H^{q}(R, F) ≅ H^{q}(R, i_{∗}i^{∗}F) ≅ H_{comp}^{q}(R, i_{∗}i^{∗}F)
= Ext_{S_{R}}^{q}(i_{∗}ℤ, i_{∗}i^{∗}F) ≅ Ext_{S_{k}}^{q}(ℤ, i^{∗}F) = H^{q}(k, i^{∗}F),

which we can alternatively write as

H^{q}(R, (M, N, φ)) ≅ H^{q}(R, (0, N, 0)) ≅ H_{comp}^{q}(R, (0, N, 0))

= Ext_{S_{R}}^{q}((0, ℤ, 0), (0, N, 0)) ≅ Ext_{S_{k}}^{q}(ℤ, N) = H^{q}(k, N).

Proof. Consider again the canonical exact sequence (27.1)

0 → j_{!}j^{∗}F → F → i_{∗}i^{∗}F → 0.

By Proposition 27.4, we have H^{q}(R, j_{!}(j^{∗}F)) = 0 for all q, so

H^{q}(R, F) ≅ H^{q}(R, i_{∗}i^{∗}F)

for all q. Now apply the relative cohomology exact sequence (Proposition 27.3) to the sheaf i_{∗}i^{∗}F, to obtain an exact sequence

⋯ → H_{comp}^{q}(R, i_{∗}i^{∗}F) → H^{q}(R, i_{∗}i^{∗}F) → H^{q}(K, j^{∗}i_{∗}i^{∗}F) → ⋯

We have

j^{∗}i_{∗}i^{∗}F = j^{∗}i_{∗}N = j^{∗}(0, N, 0) = 0.

Thus for all q,

H^{q}(R, i_{∗}i^{∗}F) ≅ H_{comp}^{q}(R, i_{∗}i^{∗}F).

By Proposition 27.5, we have

H_{comp}^{q}(R, i_{∗}i^{∗}F) ≅ H^{q}(k, i^{∗}F) = H^{q}(k, N).

Thus combining the above we get that for all q,

H^{q}(R, F) ≅ H^{q}(R, i_{∗}i^{∗}F) ≅ H_{comp}^{q}(R, i_{∗}i^{∗}F) ≅ H^{q}(k, N),

as claimed.

Remark 27.7. If i^{∗} preserves injectives, then Theorem 27.6 would follow more easily, by comparing two δ functors. Maybe i^{∗} does preserve injectives – Mazur sort of claims this on [Maz73, pg. 529]. Note that i_{∗} is merely left adjoint to i^{!}, not i^{∗}. Actually, if i^{∗} were right adjoint to a functor, that functor would probably have to be something like N ↦ (N, N, id), where G_{K} acts on the second factor through the natural homomorphism G_{K} ≅ G_{v} → G_{v}/I_{v} ≅ G_{k}. The functor c : N ↦ (N, N, id) does seem to be an exact functor from S_{k} to S_{R}, and it seems that

Hom(c(A), B) = Hom(A, i^{∗}B),

which would show that i^{∗} preserves injectives. This would thus also prove Theorem 27.6.

We have thus mastered étale cohomology over a DVR, at least assuming we know everything about Galois cohomology...

28 The Multiplicative Group over a DVR

Consider the element 𝔾_{m,R} ∈ S_{R} given by

U ↪ K̄^{∗},

where U is the group of units in the ring of integers of K^{ur}, viewed as a G_{k} = G_{v}/I_{v}-module. To get a better sense for 𝔾_{m,R}, note that we have an exact sequence in S_{R}:

\xymatrix{0 \ar[r] & U \ar[r] \ar[d] & (K^{ur})^{∗} \ar[r]^{v} \ar[d] & ℤ \ar[r] \ar[d] & 0 \\ 0 \ar[r] & K̄^{∗} \ar[r] & K̄^{∗} \ar[r] & 0 \ar[r] & 0,}

which we write as

0 → 𝔾_{m,R} → j_{∗}𝔾_{m,K} → i_{∗}ℤ → 0.

Theorem 28.1. We have

H^{q}(R, 𝔾_{m,R}) = \begin{cases}R^{∗} & if q = 0 \\ 0 & otherwise. \end{cases}

We also have

H_{comp}^{q}(R, 𝔾_{m,R}) = \begin{cases}0 & if q = 0 \\ ℤ & if q = 1 \\ 0 & if q = 2 \\ ℚ/ℤ & if q = 3 \\ 0 & if q ≥ 4. \end{cases}

Proof. By Theorem 27.6, we have

H^{q}(R, 𝔾_{m,R}) ≅ H^{q}(k, U) = H^{q}(K^{ur}/K, U).

The computation of the Galois cohomology group H^{q}(k, U) then follows from local class field theory, using the exact sequence

0 → U → (K^{ur})^{∗} ⟶^{v} ℤ → 0,  (28.1)

For example, applying G_{k} = Gal(K^{ur}/K)-cohomology to the short exact sequence (28.1) we get

0 → R^{∗} → K^{∗} ↠ ℤ → H^{1}(k, U) → H^{1}(K^{ur}/K, (K^{ur})^{∗}) = 0,

where we use that the valuation map K^{∗} → ℤ is surjective. Thus

H^{1}(k, i^{∗}𝔾_{m,R}) = H^{1}(k, U) = 0.

The compact cohomology can similarly by computed by diagram chases.

Note for comparison that

H^{q}(k, k̄^{∗}) = \begin{cases}k^{∗} & if q = 0 \\ 0 & otherwise, \end{cases}

and

H^{q}(K, K̄^{∗}) = \begin{cases}K^{∗} & if q = 0, \\ 0 & if q = 1, \\ ℚ/ℤ & if q = 2, \\ 0 & if q ≥ 3. \end{cases}

29 Étale Cohomology of Abelian Varieties

Let A be an abelian variety over K. Recall that the Néron model A of A over R is a smooth commutative group scheme over the ring O_{K} of integer of K with generic fiber A such that for all smooth commutative group schemes S the natural map

A(S) → A(S_{K})

is an isomorphism (this last statement is called the Néron mapping property).

In terms of sheaves, the Néron model is simply the pushforward of the sheaf in S_{K} defined by the abelian variety A:

Proposition 29.1. We have A = j_{∗}A as sheaves on the étale site over R.

Proof. Suppose U ∈ Ét(X) is an étale morphism. Then U → X is smooth, since étale is by definition “smooth of relative dimension 0.” Then by definition (see Section 24.6),

(j_{∗}A)(U) = A(U ×_{X} Spec(K)) = A(U_{K}) ≅ A(U),

where we use the Néron mapping property.

Thus the deep theorem that Néron models exists is really the theorem that the sheaf j_{∗}A ∈ S_{R} is representable by a smooth group scheme.

Proposition 29.2. We have

H^{1}(R, A) ≅ H^{1}(K^{ur}/K, A(K^{ur})).

Proof. By Theorem 27.6, we have

H^{1}(R, A) ≅ H^{1}(k, A(R)) ≅ H^{1}(K^{ur}/K, A(K^{ur}))

by the Néron mapping property and identification of G_{k} and G_{v}/I_{v}.

Taking special fibers, we have a canonical “connected-étale” exact sequence

0 → A_{k}^{0} → A_{k} → Φ_{A} → 0,

where Φ_{A} is the component group of A_{k}. In fact, one can show that

H^{1}(R, A) ≅ H^{1}(k, Φ_{A}),

which has the same order as Φ_{A}(k).

Next suppose A is an abelian variety over ℚ (say), Mazur does computations with H^{1}(ℤ, A) in the appendix to Rational Points on Abelian Varieties with values in Towers of Number Fields. Let

Σ = Ker (H^{1}(ℚ, A) → ⨁_{p<∞} H^{1}(ℚ_{p}, A)) ,

so � ⊂ Σ and the quotient Σ/� is a 2-group. We have an exact sequence

0 → A^{0} → A → Φ_{A} → 0,

where Φ_{A} is the component group of A (at all primes at once).

Theorem 29.3 (Mazur). The natural map i : H^{1}(ℤ, A) → H^{1}(ℚ, A) is an inclusion, and i sends the image of H^{1}(ℤ, A^{0}) isomorphically onto Σ.

Thus we have a long exact sequence

0 → Σ → H^{1}(ℤ, A) → H^{1}(ℤ, Φ_{A}) → H^{2}(ℤ, A^{0}) → …

The upshot is that H^{1}(ℤ, A) is basically � (A/ℚ), up to a power of 2 and Tamagawa numbers. It is thus an object that combines � and something involving component groups.

References

[AW67] M. F. Atiyah and C. T. C. Wall, Cohomology of groups, Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), Thompson, Washington, D.C., 1967, pp. 94–115. MR MR0219512 (36 #2593)

[Maz73] Barry Mazur, Notes on étale cohomology of number fields, Ann. Sci. École Norm. Sup. (4) 6 (1973), 521–552 (1974). MR MR0344254 (49 #8993)

[Mil] J. S. Milne, Lectures on Étale Cohomology.

[Mil80] , Étale cohomology, Princeton University Press, Princeton, N.J., 1980. MR 81j:14002

[Ser67] J-P. Serre, Local class field theory, Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), Thompson, Washington, D.C., 1967, pp. 128–161.

[Ser97] , Galois cohomology, Springer-Verlag, Berlin, 1997, Translated from the French by Patrick Ion and revised by the author.

[Sil92] J. H. Silverman, The arithmetic of elliptic curves, Springer-Verlag, New York, 1992, Corrected reprint of the 1986 original.