Étale groupoid algebras with coefficients in a sheaf and skew inverse semigroup rings. (7th December 2021)

• Record Type: Journal Article
• Authors: Gonçalves, Daniel; Steinberg, Benjamin
• Is Part Of: Canadian journal of mathematics. Volume 73:Number 6(2021)
• Page Start: 1592
• Page End: 1626
• Publication Date: 2021-12-07
• Subjects: 22A22 -- 20M18 -- 18B40 -- 16S36; Étale groupoids -- inverse semigroups -- rings; Mathematics -- Periodicals; Mathematics; Electronic journals; Periodicals; 510
• DOI: 10.4153/S0008414X20000619
• Languages: English
• ISSNs: 0008-414X

Abstract

Abstract: Given an action ${\varphi }$ of inverse semigroup S on a ring A (with domain of ${\varphi }(s)$ denoted by $D_{s^*}$ ), we show that if the ideals $D_e$, with e an idempotent, are unital, then the skew inverse semigroup ring $A\rtimes S$ can be realized as the convolution algebra of an ample groupoid with coefficients in a sheaf of (unital) rings. Conversely, we show that the convolution algebra of an ample groupoid with coefficients in a sheaf of rings is isomorphic to a skew inverse semigroup ring of this sort. We recover known results in the literature for Steinberg algebras over a field as special cases.