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Farrell-Tate cohomology

Context

Cohomology

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Special and general types

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Special notions

Variants

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  • differential cohomology

Extra structure

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Operations

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Theorems

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Representation theory

Ingredients

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Definitions

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Geometric representation theory

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Theorems

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

  • H^ n(G,A)=H n(G,A)\hat{H}^n (G, A) = H^n(G, A), for n1n \geq 1.
  • H^ 0(G,A)=cokerN\hat{H}^0 (G, A) = coker N, for n=0n = 0.
  • H^ 1(G,A)=kerN\hat{H}^{-1} (G, A) = ker N, for n=0n = 0.
  • H^ n(G,A)=H (n+1)(G,A)\hat{H}^n (G, A) = H_{-(n+1)}(G, A), for n2n \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 GG and any naive GG-spectrum EE.

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

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