nLab
Farrell-Tate cohomology

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Representation theory

Contents

Idea

What is called Tate cohomology are cohomology groups H^(G,N)\widehat{H}(G,N) associated to a representation NN of a finite group GG. In terms of the Tate spectrum HN tGH N^{t G} of the Eilenberg-MacLane spectrum HNH N of NN, these may be expressed as its stable homotopy groups:

H^ n(G,N)π n(HN tG). \widehat H^{-n}(G,N) \simeq \pi_n( H N^{t G}) \,.

(e.g. Nikolaus-Scholze 17, p. 13)

What is called (generalized) Farrell-Tate cohomology is a generalization of this construction to possibly infinite discrete groups and topological groups GG.

(e.g. Klein 02, Nikolaus-Scholze 17, section I.4)

History of the idea

Tate cohomology was introduced in Tate52 by John Tate for the purposes of class field theory. When a finite group GG acts on an abelian group AA, then there is a natural ‘norm’ map NN from H 0(G,A)H_0(G, A) to H 0(G,A)H^0(G,A), a ggaa \mapsto \sum_g g a.

Then the Tate cohomology groups are

In (Farrell78), Farrell generalized this construction to possibly infinite discrete groups of finite virtual cohomological dimension.

Later this was generalized further in (Klein 02) to any topological (or discrete) group GG and any naive GG-spectrum EE.

References