Smoothness of stabilisers in generic characteristic
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arXiv
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| Format: | Preprint |
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2018
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| author | Martin, Benjamin Stewart, David I. Topley, Lewis |
| author_facet | Martin, Benjamin Stewart, David I. Topley, Lewis |
| contents | Let $R$ be a commutative unital ring. Given a finitely presented affine $R$-group scheme $G$ acting on a separated scheme $X$ of finite type over $R$, we show that there is a prime $p_0$ such that for any $R$-algebra $k$ which is an algebraically closed field of characteristic $p\geq p_0$, the centraliser in $G_k$ of any closed subscheme of $X_k$ is smooth. When $X$ is not necessarily separated we show similarly that for any closed subscheme $Y \subseteq X$ there is a $p_1$ depending on $Y$ such that when $k$ has characteristic $p \geq p_1$ the normaliser of $Y$ in $G_k$ is smooth. We prove these results using the Lefschetz principle together with careful application of Gröbner basis techniques, and using a suitable notion of the complexity of an action.
We apply our results to demonstrate that the Kostant-Kirillov-Souriau theorem holds for Lie algebras of algebraic groups in large positive characteristics. In particular, every such Lie algebra decomposes as a disjoint union of symplectic varieties, each of which is a coadjoint orbit. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1810_12628 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Smoothness of stabilisers in generic characteristic Martin, Benjamin Stewart, David I. Topley, Lewis Group Theory Commutative Algebra Algebraic Geometry Representation Theory Let $R$ be a commutative unital ring. Given a finitely presented affine $R$-group scheme $G$ acting on a separated scheme $X$ of finite type over $R$, we show that there is a prime $p_0$ such that for any $R$-algebra $k$ which is an algebraically closed field of characteristic $p\geq p_0$, the centraliser in $G_k$ of any closed subscheme of $X_k$ is smooth. When $X$ is not necessarily separated we show similarly that for any closed subscheme $Y \subseteq X$ there is a $p_1$ depending on $Y$ such that when $k$ has characteristic $p \geq p_1$ the normaliser of $Y$ in $G_k$ is smooth. We prove these results using the Lefschetz principle together with careful application of Gröbner basis techniques, and using a suitable notion of the complexity of an action. We apply our results to demonstrate that the Kostant-Kirillov-Souriau theorem holds for Lie algebras of algebraic groups in large positive characteristics. In particular, every such Lie algebra decomposes as a disjoint union of symplectic varieties, each of which is a coadjoint orbit. |
| title | Smoothness of stabilisers in generic characteristic |
| topic | Group Theory Commutative Algebra Algebraic Geometry Representation Theory |
| url | https://arxiv.org/abs/1810.12628 |