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
are defined by where by the "obvious" fixed point we mean those of the form
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: Suppose that A is a module over a finite group G and a is an element of
, 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.
F. Thomas Farrell extended Tate cohomology groups to the case of all groups G of finite virtual cohomological dimension.
are isomorphic to the usual cohomology groups whenever n is greater than the virtual cohomological dimension of the group G. Finite groups have virtual cohomological dimension 0, and in this case Farrell's cohomology groups are the same as those of Tate.