## Homology, Homotopy and Applications

- Homology Homotopy Appl.
- Volume 7, Number 1 (2005), 87-108.

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

#### Abstract

It is proved that the homology and cohomology theories of groups and associative algebras are non-abelian derived functors of the cokernel and kernel groups of higher dimensions of their defining standard chain and cochain complexes respectively. The same results are also obtained for the relative (co)homology of groups, the mod q cohomology of groups and the cohomology of groups with operators. This allowed us to give an alternative approach to higher Hopf formulas for integral homology of groups. An axiomatic characterization of the relative cohomology of groups is given and higher relative $(n+1)$-th cohomology of groups is described in terms of n-fold extensions.

#### Article information

**Source**

Homology Homotopy Appl., Volume 7, Number 1 (2005), 87-108.

**Dates**

First available in Project Euclid: 13 February 2006

**Permanent link to this document**

https://projecteuclid.org/euclid.hha/1139839508

**Mathematical Reviews number (MathSciNet)**

MR2155520

**Zentralblatt MATH identifier**

1080.18008

**Subjects**

Primary: 18G10: Resolutions; derived functors [See also 13D02, 16E05, 18E25]

Secondary: 18G25: Relative homological algebra, projective classes 18G50: Nonabelian homological algebra 18G55: Homotopical algebra

#### Citation

Inassaridze, Hvedri. More about the (co)homology of groups and associative algebras. Homology Homotopy Appl. 7 (2005), no. 1, 87--108. https://projecteuclid.org/euclid.hha/1139839508