Tate Cohomology Group - Tate's Theorem

Tate's Theorem

Tate's theorem (Tate 1952) gives conditions for multiplication by a cohomology class to be an isomorphism between cohomology groups. There are several slightly different versions of it; a version that is particularly convenient for class field theory is as follows:

Suppose that A is a module over a finite group G and a is an element of H2(G,A), such that for every subgroup E of G

• H1(E,A) is trivial, and
• H2(E,A) is generated by Res(a) which has order E.

Then cup product with a is an isomorphism

for all n; in other words the graded Tate cohomology of A is isomorphic to the Tate cohomology with integral coefficients, with the degree shifted by 2.