Homology and cohomology via the partial group algebra

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 $c{d}_{\mathbb{K}}(G)$ (over a field $\mathbb{K}$) with the partial cohomological dimension $c{d}_{\mathbb{K}}^{par}(G)$ (over $\mathbb{K}$) and show that $c{d}_{\mathbb{K}}^{par}(G)\ge c{d}_{\mathbb{K}}(G)$ and that there is equality for $G=\mathbb{Z}$.

arXiv commentThe statement of Corollary C was changed and Section 6 was basically rewritten as several of the results there were not correct. Section 7 was removed. Some other small changes were made through the text