Geodesic
Étale groupoid algebras with coefficients in a sheaf and skew inverse semigroup rings
Canadian journal of mathematics, Tome 73 (2021) no. 6, pp. 1592-1626

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DOI

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.
DOI : 10.4153/S0008414X20000619
Mots-clés : Étale groupoids, inverse semigroups, rings 
Gonçalves, Daniel; Steinberg, Benjamin. Étale groupoid algebras with coefficients in a sheaf and skew inverse semigroup rings. Canadian journal of mathematics, Tome 73 (2021) no. 6, pp. 1592-1626. doi: 10.4153/S0008414X20000619
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     title = {\'Etale groupoid algebras with coefficients in a sheaf and skew inverse semigroup rings},
     journal = {Canadian journal of mathematics},
     pages = {1592--1626},
     year = {2021},
     volume = {73},
     number = {6},
     doi = {10.4153/S0008414X20000619},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/S0008414X20000619/}
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