# 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 Template:Harvs, and are used in class field theory.

## Contents

## Definition

If *G* is a finite group and *A* a *G*-module, then there is a natural map *N* from *H*_{0}(*G*,*A*) to
*H*^{0}(*G*,*A*) taking a representative *a* to Σ *g*(*a*) (the sum over all *G*-conjugates of *a*). The **Tate cohomology groups** are defined by

- for
*n*≥ 1. - quotient of
*H*^{0}(*G*,*A*) by normsTemplate:Clarify - quotient of norm 0 elements of
*H*^{0}(*G*,*A*) by principal norm 0 elementsTemplate:Clarify - for
*n*≤ −2.

## 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 Template:Harv 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 *H*^{2}(*G*,*A*), such that for every subgroup *E* of *G*

*H*^{1}(*E*,*A*) is trivial, and*H*^{2}(*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.

## 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
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.

## See also

## References

- M. F. Atiyah and C. T. C. Wall, "Cohomology of Groups", in
*Algebraic Number Theory*by J. W. S. Cassels, A. Frohlich ISBN 0-12-163251-2, Chapter IV. See section 6. - Kenneth S. Brown,
*Cohomology of Groups*, ISBN 0-387-90688-6 - Farrell, F. Thomas
*An extension of Tate cohomology to a class of infinite groups.*J. Pure Appl. Algebra 10 (1977/78), no. 2, 153-161. - {{#invoke:citation/CS1|citation

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