Categorical quotients for actions of groupoids on varieties
Ian M. Musson
TL;DR
The paper addresses quotients by groupoid actions on varieties, focusing on Weyl groupoids acting on an affine variety $X$ and the quotient map $\pi: X \to X//\mathfrak{G}=\operatorname{Spec}\mathcal{O}(X)^{\mathfrak{G}}$. It develops a framework that extends the Mumford-Fogarty-Kirwan universality results to quotients by groupoids, leveraging the Jacobson property and the role of $k$-rational points. The main contributions show that for uncountable $k$, the quotient $\pi$ is a geometric quotient and universal in the category of $k$-schemes, both in the affine case and for Weyl groupoid actions $\mathfrak{W}^c$ and $\mathfrak{W}_*^c$, with the invariant rings corresponding to centers $Z(\mathfrak{g})$ and to supercharacter algebras, respectively. The work also clarifies when universality can be expected and raises questions about relaxing the uncountability assumption and the weak Nullstellensatz in more general, possibly non-finitely generated, settings.
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 $π: X \longrightarrow Y= {\operatorname{Spec }\;} \mathcal{O}(X)^\mathfrak{W}$ 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 $π$ 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.