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differential cohomology
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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:
(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)
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
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 $E$.
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
Last revised on July 24, 2017 at 06:46:50. See the history of this page for a list of all contributions to it.