Exercise Set 4: Tate’s Cohomology Groups

Math 582e, Winter 2010, University of Washington

Due Monday, February 8, 2010

1. Let G be a finite group and A a co-induced module. In class we proved that Ĥ^{0}(G, A) = 0. Prove that Ĥ_{0}(G, A) = 0 as well.

2. Prove that H_{1}(G, ℤ) = G/G′, where G′ is the commutator subgroup of G. This is proved in Serre’s Local Fields book, but there may be gaps in the argument, so fill them in.