Categorical quotients for actions of groupoids on varieties

TL;DR

This paper proves that for certain groupoid actions on affine varieties over uncountable fields, the quotient map is a universal geometric quotient, extending classical results to more general groupoid actions.

Contribution

It extends the concept of geometric quotients to actions of groupoids on varieties, demonstrating universality in the category of schemes.

Findings

The quotient map is a geometric quotient for uncountable fields.

The quotient map is universal in the category of schemes.

Extension of classical quotient results to groupoid actions.

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.