Homology and cohomology via the partial group algebra

TL;DR

This paper explores the relationship between partial homology and cohomology of groups via the partial group algebra, linking it to classical group (co)homology and comparing their cohomological dimensions.

Contribution

It introduces a ring-theoretic approach to partial (co)homology using the partial group algebra and compares partial and classical cohomological dimensions.

Findings

Partial homology and cohomology relate to classical group (co)homology with non-trivial coefficients.

The partial cohomological dimension is greater than or equal to the classical dimension.

Equality of these dimensions holds for the infinite cyclic group Z.

Abstract

We study partial homology and cohomology from ring theoretic point of view via the partial group algebra $\mathbb{K}_{par}G$. In particular, we link the partial homology and cohomology of a group $G$ with coefficients in an irreducible (resp. indecomposable) $\mathbb{K}_{par}G$-module with the ordinary homology and cohomology groups of $G$ with in general non-trivial coefficients. Furthermore, we compare the standard cohomological dimension $cd_{ \ \mathbb{K}}(G)$ (over a field $\mathbb{K}$) with the partial cohomological dimension $cd_{ \ \mathbb{K}}^{par}(G)$ (over $\mathbb{K}$) and show that $cd_{ \ \mathbb{K}}^{par}(G) \geq cd_{ \ \mathbb{K}}(G)$ and that there is equality for $G = \mathbb{Z}$.