In mathematics, Tate cohomology groups are a slightly modified form of the usual cohomology groups of a finite group that combine homology and cohomology groups into one sequence. They were introduced by John Tate (1952, p. 297), and are used in class field theory.
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Definition
If G is a finite group and A a G-module, then there is a natural map N from H0(G,A) to H0(G,A) taking a representative a to Σga (the sum over all G-conjugates of a). The Tate cohomology groups
Properties
If
is a short exact sequence of G-modules, then we get the usual long exact sequence of Tate cohomology groups:
If A is an induced G module then all Tate cohomology groups of A vanish.
The zeroth Tate cohomology group of A is
(Fixed points of G on A)/(Obvious fixed points of G acting on A)where by the "obvious" fixed point we mean those of the form Σ g(a). In other words, the zeroth cohomology group in some sense describes the non-obvious fixed points of G acting on A.
The Tate cohomology groups are characterized by the three properties above.
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
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.
Tate-Farrell cohomology
Farrell extended Tate cohomology groups to the case of all groups G of finite virtual cohomological dimension. In Farrell's theory, the groups