Homomorphisms of algebraic groups: representability and rigidity

Autor:	Brion, Michel
Rok vydání:	2021
Předmět:	Mathematics - Algebraic Geometry 14L15 (Primary) 14D20 14K05 14L30 20G15 (Secundary)
Druh dokumentu:	Working Paper
Popis:	Given two algebraic groups $G$, $H$ over a field $k$, we investigate the representability of the functor of morphisms (of schemes) $\mathbf{Hom}(G,H)$ and the subfunctor of homomorphisms (of algebraic groups) $\mathbf{Hom}_{\rm gp}(G,H)$. We show that $\mathbf{Hom}(G,H)$ is represented by a group scheme, locally of finite type, if the $k$-vector space $\mathcal{O}(G)$ is finite-dimensional; the converse holds if $H$ is not \'etale. When $G$ is linearly reductive and $H$ is smooth, we show that $\mathbf{Hom}_{\rm gp}(G,H)$ is represented by a smooth scheme $M$; moreover, every orbit of $H$ acting by conjugation on $M$ is open. Comment: Minor changes. Final version, accepted for publication in a volume of Michigan Mathematical Journal dedicated to Gopal Prasad
Databáze:	arXiv
Externí odkaz:	http://arxiv.org/abs/2101.12460 Zobrazit plný text záznamu View this record from Arxiv