The étale groupoid of an inverse semigroup as a groupoid of filters. (Q2861578)

As the authors write in the abstract: ``Paterson showed how to construct an étale groupoid from an inverse semigroup using ideas from functional analysis [see \textit{A. L. T. Paterson}, Groupoids, inverse semigroups, and their operator algebras. Boston: Birkhäuser (1999; Zbl 0913.22001)]. This construction was later simplified by \textit{D. H. Lenz}, [Proc. Edinb. Math. Soc., II. Ser. 51, No. 2, 387-406 (2008; Zbl 1153.22003)]. We show that Lenz's construction can itself be further simplified by using filters: the topological groupoid associated with an inverse semigroup is precisely a groupoid of filters.''NEWLINENEWLINE Let \(A\) be a ring and \(M\) unitary left \(A\)-module, that is, \(AM=M\). In particular, a simple \(A\)-module is an \(A\)-module \(M\) such that \(AM\neq 0\) and there are no nonzero proper submodules of \(M\). If \(e\) is an idempotent of \(A\) and \(M\) is an \(A\)-module, then \(eM\) is an \(eAe\)-module. For an \(eAe\)-module \(N\), define \(Ind_e(N)=Ae\otimes_{eAe}N\). If the \(\mathcal D\)-class of \(e\) contains only finitely many idempotents then we say that \(e\) has \textit{finite index} in \(S\).NEWLINENEWLINE The main result of paper is the following theorem. Let \(k\) be a field and \(S\) an inverse semigroup. Then the finite-dimensional simple \(kS\)-modules are precisely those of the form \(Ind_e(N)\) where \(e\) is a finite-index idempotent of \(\mathcal L(S)\) and \(N\) is a finite-dimensional simple \(kG_e\)-module.