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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Online-Zugang: | https://arxiv.org/abs/2501.13510 |
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| _version_ | 1866910795512676352 |
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| author | Musson, Ian M. |
| author_facet | Musson, Ian M. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_13510 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Categorical quotients for actions of groupoids on varieties Musson, Ian M. Algebraic Geometry Representation Theory 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. |
| title | Categorical quotients for actions of groupoids on varieties |
| topic | Algebraic Geometry Representation Theory |
| url | https://arxiv.org/abs/2501.13510 |