Homology and cohomology via the partial group algebra

Author(s)

Alves, Marcelo Muniz ; Dokuchaev, Mikhailo ; Kochloukova, Dessislava H.

• Document type: Journal article
• Source: QUARTERLY JOURNAL OF MATHEMATICS; v. 75, n. 2, p. 49-pg., 2024-05-02.

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 $G = \mathbb{Z}$. (AU)

• FAPESP's process: 20/16594-0 - Non commutative rings and applications
• Grantee: Francisco Cesar Polcino Milies
• Support Opportunities: Research Projects - Thematic Grants
• FAPESP's process: 18/23690-6 - Structures, representations, and applications of algebraic systems
• Grantee: Ivan Chestakov
• Support Opportunities: Research Projects - Thematic Grants