On the homology of partial group representations
Emmanuel Jerez
TL;DR
The paper establishes a precise bridge between partial group (co)homology and classical group (co)homology by introducing the universal globalization $\Lambda(M)$ of partial representations. It proves $H^{par}_{\bullet}(G,M)\cong H_{\bullet}(G,\Lambda(M))$ and constructs a cohomology spectral sequence with $E_{2}^{p,q}=H^{p}(G,\operatorname{Ext}^{q}_{K_{par}}(\Lambda(K_{par}),M))$ that converges to $H^{par}_{\bullet}(G,M)$; under countable or finite conditions, the spectral sequence collapses to explicit isomorphisms such as $H^{\bullet}_{par}(G,M)\cong H^{\bullet}(G,\operatorname{Hom}_{K_{par}}(\Lambda(K_{par}),M))$. The globalization machinery, including the partial tensor product and the globalization functor $\Lambda$, enables transferring computations from partial to global settings, with applications to Shapiro's lemma and the Lyndon–Hochschild–Serre framework for partial actions. The results yield practical toolsets for deriving classical invariants from partial data and confirm conjectures in the countable case, significantly expanding the computational and theoretical reach of partial (co)homology.
Abstract
We study how the partial group (co)homology of a group $G$ with coefficient in a partial representation $M$ can be described using the usual group (co)homology. To address this, we introduce the concept of the \textit{universal globalization} $Λ(M)$ of a partial group representation $M$ of $G$. Our main result shows that the partial group homology $H^{\text{par}}_{\bullet}(G, M)$ is naturally isomorphic to the classical group homology $H_{\bullet}(G, Λ(M))$. We extend this result to the cohomological framework, obtaining a spectral sequence involving the classical group cohomology that converges to the partial group cohomology. Notably, when $G$ is countable, the spectral sequence collapses, resulting in a natural isomorphism $H^{\bullet}_{\text{par}}(G, M) \cong H^{\bullet}(G, \operatorname{Hom}_{K_{\text{par}} G}(Λ(K_{par}G), M))$, where $K_{par}G$ stands for the partial group algebra of $G$.