nLab Farrell-Tate cohomology

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