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First of all I want to say that I am not at all an expert in Group cohomology . Recently I attended a seminar where the speaker mentioned about something called Tate cohomology groups which in someway relate group homology and group cohomology in one sequence.

My question is the following:

Is there any result towards the characterization of the finite groups $G$ and $G$-modules $A$ such that the natural map $N:H_0(G,A) \rightarrow H^{0}(G,A)$ appearing in Tate cohomology is an isomorphism of groups?

(My reference is https://en.wikipedia.org/wiki/Tate_cohomology_group)

It seems interesting to me because when $N$ is an isomorphism it represents some sort of notion of duality between group homology and group cohomology at the zeroth level.

I asked this question to the speaker but did not get satisfactory answer . So I am asking here for an answer / partial answer to my question.

I apologise in advance if my question sounds stupid.

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    $\begingroup$ It holds, for example, if A is a projective Z[G]-module (III.1.1(c) in Brown's book 'Cohomology of Groups') $\endgroup$
    – Drew Heard
    Commented Aug 4, 2020 at 6:55
  • $\begingroup$ @DrewHeard Thanks for the Reference. I guess you mean Chapter 3, Section 1 in Brown's book. I have the first edition of the book with me but I could not find 1(c) in section 1 of chapter 3 there. Is it a different edition? $\endgroup$
    – Adittya Chaudhuri
    Commented Aug 4, 2020 at 7:14
  • $\begingroup$ @DrewHeard Thanks I found it. It was in the Exercise. $\endgroup$
    – Adittya Chaudhuri
    Commented Aug 4, 2020 at 7:35

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