Restriction and Extension of Partial Actions

TL;DR

This paper explores conditions under which partial actions of isotropy groups on ideals can be extended to partial actions of the entire groupoid, with implications for Morita and Galois theories.

Contribution

It investigates the extension of partial group actions to partial groupoid actions and addresses the globalization problem with applications to algebraic theories.

Findings

Conditions for extending partial group actions to groupoid actions

Results on the globalization of partial actions

Applications to Morita and Galois theories

Abstract

Given a partial action $\alpha=(A_g,\alpha_g)_{g\in \mathcal{G}}$ of a connected groupoid $\mathcal{G}$ on a ring $A$ and an object $x$ of $\mathcal{G}$, the isotropy group $\mathcal{G}(x)$ acts partially on the ideal $A_x$ of $A$ by the restriction of $\alpha$. In this paper we investigate the following reverse question: under what conditions a partial group action of $\mathcal{G}(x)$ on an ideal of $A$ can be extended to a partial groupoid action of $\mathcal{G}$ on $A$? The globalization problem and some applications to the Morita and Galois theories are also considered, as extensions of similar results from the group actions case.