Modq cohomology and Tate–Vogel cohomology of groups

Abstract

The notions of mod q cohomology and Tate-Farrell-Vogel cohomology of groups are introduced, where q is a positive integer. The ÿrst and the second mod q cohomology groups are described in terms of torsors and extensions respectively. The mod q cohomology of groups is expressed as cotriple cohomology. The reduction of mod q Tate-Farrell-Vogel cohomology theory to the case q = p m with p a prime is shown. For ÿnite groups with periodic cohomology the periodicity of mod q Tate cohomology is proved.

More about the (co)homology of groups and associative algebras

“…This case has been already investigated in [10], Section 2. However, in this context it is an important example showing the necessity to use the non-abelian derived functors of higher dimensional kernels of the defining cochain complex to express the corresponding cohomology groups as cotriple cohomology.…”

“…Our aim is to develop the observation that the non-abelian derived functors of the groups of cokernel and kernel of higher dimensions of the above mentioned chain and cochain complexes respectively could be also used to describe the (co)homology of groups and associative algebras. In this context we shall consider the classical Eilenberg-MacLane (co)homology of groups (Theorems 1 and 3), the Hochschild (co)homology of associative algebras (Theorem 16), the relative cohomology of groups (Theorem 8), the mod q cohomology of groups defined in [9,10 ] (Theorem 6) and the cohomology of groups with operators defined in [7] (Theorem 15). This approach to cotriple (co)homology theory is particularly needed to the investigation of the mod q cohomology theory of groups (see [10]).…”

“…In this context we shall consider the classical Eilenberg-MacLane (co)homology of groups (Theorems 1 and 3), the Hochschild (co)homology of associative algebras (Theorem 16), the relative cohomology of groups (Theorem 8), the mod q cohomology of groups defined in [9,10 ] (Theorem 6) and the cohomology of groups with operators defined in [7] (Theorem 15). This approach to cotriple (co)homology theory is particularly needed to the investigation of the mod q cohomology theory of groups (see [10]). Moreover these results allow us to give an alternative purely group theoretic description of the integral homology of groups (Theorem 4).…”

INTERNAL q-HOMOLOGY OF CROSSED MODULES

“…Example 2.5. If A is a G-module, the q-centre of (A, G, 0) is (H 0 (G, A; Z/qZ), Z q (G)∩ st G (A), 0), where H 0 (G, A; Z/qZ) is the zeroth mod q cohomology group introduced in [4].…”

Finite K-theory spaces

“…In [15], Karoubi and Lambre introduced the Hochschild homology with finite coefficients, constructed the Dennis trace map from Hochschild homology with finite coefficients and found an unexpected relationship with number theory. The mod m Tate-Farell-Vogel cohomology of groups has been introduced in [11] having applications to mod m algebraic K-theory.…”