Categorical quotients for actions of groupoids on varieties

Abstract

For certain actions of the Weyl groupoid $\mathfrak{W}$ from [Sergeev and Veselov, Grothendieck rings of basic classical Lie superalgebras, Ann Math, 2011] on an affine variety $X$, geometric properties of the map $\pi: X \longrightarrow Y= {\operatorname{Spec }\;} \mathcal{O}(X)^\mathfrak{W}$ were studied in [Musson, On the geometry of some algebras related to the Weyl groupoid, Contemp. Math. 2024], In this paper we show that if the base field ${\mathtt k}$ is uncountable, the map $\pi$ is a geometric quotient which is universal in the category of ${\mathtt k}$-schemes. To do this we adapt a result from [{Mumford}, {Fogarty}, {Kirwan}, {1994}], showing that a geometric quotient is universal in the category of ${\mathtt k}$-schemes, to quotients by groupoids and more generally by equivalence relations. In our approach a key role is played by the closed points and Jacobson schemes.