Homomorphisms of algebraic groups: representability and rigidity

TL;DR

This paper studies the conditions under which the functor of morphisms and homomorphisms between algebraic groups are representable by schemes, revealing new criteria related to finiteness, reductiveness, and smoothness.

Contribution

It establishes new representability results for morphism and homomorphism functors between algebraic groups, including criteria involving finite-dimensionality and properties like reductiveness and smoothness.

Findings

Hom functor is represented by a group scheme if $ ext{dim}_k ext{O}(G)$ is finite.

Converse holds if $H$ is not étale.

Hom_{gp}(G,H) is smooth when $G$ is linearly reductive and $H$ is smooth.

Abstract

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.