Separability for relative extensions of object unital strongly groupoid graded rings

TL;DR

This paper characterizes when a ring graded by a groupoid extension is separable, generalizing several classical results and applying to various algebraic structures like crossed products and matrix rings.

Contribution

It provides a new necessary and sufficient condition for separability in object unital strongly groupoid graded rings, unifying and extending previous theorems.

Findings

Generalizes classical separability theorems for matrix and group-graded rings.

Provides explicit trace-based criteria for separability in groupoid-graded rings.

Applies to a wide class of algebraic structures including crossed products and matrix rings.

Abstract

We prove that if $R$ is a ring that is object unital and strongly graded by a groupoid $\Gamma$, and if $\Delta$ is a wide subgroupoid of $\Gamma$, then $R/R_\Delta$ is separable if and only if, for each $e \in \Gamma_0$, there exist $f \in [e]$ and $r \in C_{R_0}(R_\Lambda) := \{ x \in R_0 \mid xy = yx \text{ for all } y \in R_\Lambda \}$ with ${\rm tr}_{\Gamma/\Delta}^f(r) = 1_{R_f}$. Here, $\Gamma_0$ denotes the set of objects of $\Gamma$, $[e]$ the connected component of $\Gamma_0$ containing $e$, $\Lambda$ the isotropy groupoid of $\Delta$, and ${\rm tr}_{\Gamma/\Delta}^f$ the relative trace map at $f$. This result simultaneously generalizes earlier theorems on separability for matrix rings and group-graded rings due to DeMeyer-Ingraham, N{\v a}st{\v a}sescu, Van den Bergh, Van Oystaeyen, Miyashita, Theohari-Apostolidi, and Vavatsoulas, as well as results on groupoid-graded rings due to Cala, Lundstr\"{o}m, and Pinedo. As an application, we consider separability for object crossed products, including object twisted groupoid rings, classical groupoid rings and matrix rings, as well as crossed product algebras defined by infinite separable field extensions.