group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
geometric representation theory
representation, 2-representation, ∞-representation
Grothendieck group, lambda-ring, symmetric function, formal group
principal bundle, torsor, vector bundle, Atiyah Lie algebroid
Eilenberg-Moore category, algebra over an operad, actegory, crossed module
Be?linson-Bernstein localization?
What is called Tate cohomology are cohomology groups associated to a representation of a finite group . In terms of the Tate spectrum of the Eilenberg-MacLane spectrum of , 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 .
(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 acts on an abelian group , then there is a natural ‘norm’ map from to , .
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 and any naive -spectrum .
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 10:46:50. See the history of this page for a list of all contributions to it.