# Globalization of group cohomology in the sense of Alvares-Alves-Redondo

## TL;DR

This paper explores the globalization of a new group cohomology theory introduced by Alvares, Alves, and Redondo, linking it to classical cohomology via algebraic globalization and cocycle extendibility.

## Contribution

It establishes conditions under which the new cohomology theory can be globalized and relates it to classical cohomology groups through multiplier algebras.

## Key findings

- Cocycles are globalizable when the algebra is a product of blocks.
- Globalizations of cohomologous cocycles remain cohomologous.
- The new cohomology group is isomorphic to a classical cohomology group with multiplier algebra coefficients.

## Abstract

Recently E. R. Alvares, M. M. Alves and M. J. Redondo introduced a cohomology for a group $G$ with values in a module over the partial group algebra $K_{\mathrm{par}}(G)$, which is different from the partial group cohomology defined earlier by the first two named authors of the present paper. Given a unital partial action $\alpha$ of $G$ on a (unital) algebra $\mathcal{A}$ we consider $\mathcal{A}$ as a $K_{\mathrm{par}}(G)$-module in a natural way and study the globalization problem for the cohomology in the sense of Alvares-Alves-Redondo with values in $\mathcal{A}$. The problem is reduced to an extendibility property of cocycles. Furthermore, assuming that $\mathcal{A}$ is a product of blocks, we prove that any cocycle is globalizable, and globalizations of cohomologous cocycles are also cohomologous. As a consequence we obtain that the Alvares-Alves-Redondo cohomology group $H_{par}^n(G,\mathcal{A})$ is isomorphic to the usual cohomology group $H^n(G,\mathcal{M}(\mathcal{B}))$, where $\mathcal{M}(\mathcal{B})$ is the multiplier algebra of $\mathcal{B}$ and $\mathcal{B}$ is the algebra under the enveloping action of $\alpha$.