Give an example of a $G$-module $M$, such that $\hat{H}^{*}(G,M)=0$, but $M$ is not cohomologically trivial. Here $\hat{H}^{*}(G,-)$ means Tate Cohomology.
Find an example about cohomologically trivial
0
$\begingroup$
group-cohomology
1 Answers
1
This hint is from Serre's Local fields:
Take $G$ to be the cyclic group of order 6 and let $A = \mathbb{Z}/3\mathbb{Z}$. Let $G$ operate on $A$ by $x \mapsto -x$. Then show that $\hat{H^{0}}(H,A) \neq 0$, where $H$ is the subgroup of order 3.
Hope that helps.
-
0But H is a summand of G,does H^0(G,A)=0?About the operate on A? – 2012-02-25