Homology and cohomology via the partial group algebra (2006.10173v2)

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 $G = \mathbb{Z}$.