Étale groupoid algebras with coefficients in a sheaf and skew inverse semigroup rings. (7th December 2021) - Gonçalves, Daniel; Steinberg, Benjamin
Abstract: Given an action ${φ }$ of inverse semigroup S on a ring A (with domain of ${φ }(s)$ 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.