On the separability of the partial skew groupoid ring

TL;DR

This paper establishes necessary and sufficient conditions for the partial and global skew groupoid rings to be separable extensions of the base ring, advancing understanding of their algebraic structure.

Contribution

It provides a complete characterization of when partial and global skew groupoid rings are separable over the base ring, extending previous results to the groupoid context.

Findings

Characterization of separability conditions for partial skew groupoid rings

Extension of separability criteria from groups to groupoids

Conditions depend on the structure of the groupoid and the action

Abstract

Given a partial (resp. a global) action $\alpha$ of a connected finite groupoid $G$ on a ring $A$, we determine necessary and sufficient conditions for the partial (resp. global) skew groupoid ring $A\star_{\alpha} G$ to be a separable extension of $A$.