Tate cohomology group

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.

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 ${\displaystyle {\hat {H}}^{n}(G,A)}$ are defined by

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

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 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:

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