Separability for relative extensions of object unital strongly groupoid graded rings

Abstract

We prove that if $R$ is a ring that is object unital and strongly graded by a groupoid $Γ$, and if $Δ$ is a wide subgroupoid of $Γ$, then $R/R_Δ$ is separable if and only if, for each $e \in Γ_0$, there exist $f \in [e]$ and $r \in C_{R_0}(R_Λ) := \{ x \in R_0 \mid xy = yx \text{ for all } y \in R_Λ\}$ with ${\rm tr}_{Γ/Δ}^f(r) = 1_{R_f}$. Here, $Γ_0$ denotes the set of objects of $Γ$, $[e]$ the connected component of $Γ_0$ containing $e$, $Λ$ the isotropy groupoid of $Δ$, and ${\rm tr}_{Γ/Δ}^f$ 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, Lundström, 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.