Homology and cohomology via the partial group algebra

Abstract

ABSTRACT We study partial homology and cohomology from the ring theoretic point of view via the partial group algebra $\hspace{2pt} \mathbb{K}_{par} G$. In particular, we link the partial homology and cohomology of a group G with coefficients in an irreducible (resp. indecomposable) $\hspace{2pt} \mathbb{K}_{par} G$-module M with the ordinary homology and cohomology groups of a subgroup H of $G,$ where H depends on $M,$ with coefficients in an appropriate irreducible (resp. indecomposable) $\hspace{2pt} \mathbb{K} H$-module. Furthermore, we compare the standard cohomological dimension $cd_{\hspace{2pt} \mathbb{K}}(G)$ (over a field $\hspace{2pt} \mathbb{K}$) with the partial cohomological dimension $cd_{\hspace{2pt} \mathbb{K}}^{par}(G)$ (over $\hspace{2pt} \mathbb{K}$) and show that $cd_{\hspace{2pt} \mathbb{K}}^{par}(G) \geq cd_{\hspace{2pt} \mathbb{K}}(G)$ and that there is equality for $G = \mathbb{Z}$.