nLab
Farrell-Tate cohomology
Context
Cohomology
cohomology

Special and general types
Special notions
Variants
Operations
Theorems
Representation theory
representation theory

geometric representation theory

Ingredients
Definitions
representation , 2-representation , ∞-representation

group , ∞-group

group algebra , algebraic group , Lie algebra

vector space , n-vector space

affine space , symplectic vector space

action , ∞-action

module , equivariant object

bimodule , Morita equivalence

induced representation , Frobenius reciprocity

Hilbert space , Banach space , Fourier transform , functional analysis

orbit , coadjoint orbit , Killing form

unitary representation

geometric quantization , coherent state

socle , quiver

module algebra , comodule algebra , Hopf action , measuring

Geometric representation theory
D-module , perverse sheaf ,

Grothendieck group , lambda-ring , symmetric function , formal group

principal bundle , torsor , vector bundle , Atiyah Lie algebroid

geometric function theory , groupoidification

Eilenberg-Moore category , algebra over an operad , actegory , crossed module

reconstruction theorems

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

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

(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 $G$ .

(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 $G$ acts on an abelian group $A$ , then there is a natural ‘norm’ map $N$ from $H_0(G, A)$ to $H^0(G,A)$ , $a \mapsto \sum_g g a$ .

Then the Tate cohomology groups are

$\hat{H}^n (G, A) = H^n(G, A)$ , for $n \geq 1$ .
$\hat{H}^0 (G, A) = coker N$ , for $n = 0$ .
$\hat{H}^{-1} (G, A) = ker N$ , for $n = 0$ .
$\hat{H}^n (G, A) = H_{-(n+1)}(G, A)$ , for $n \leq -2$ .
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 $G$ and any naive $G$ -spectrum $E$ .

References
Thomas Nikolaus , Peter Scholze , section I-4 of On topological cyclic homology (arXiv:1707.01799 )

John Tate , The higher dimensional cohomology groups of class field theory , Ann. of Math. (2), 1952, 56: 294–297.

F. Thomas Farrell, An extension of Tate cohomology to a class of infinite groups , J. Pure Appl. Algebra 10 (1977/78), no. 2, 153-161.

John Klein , Axioms for generalized Farrell–Tate cohomology , 2002, pdf