Homomorphisms of algebraic groups: representability and rigidity

Michel Brion

Published 2021-01-29Version 1

Consider two group schemes $G$, $H$, locally of finite type over a field $k$. We show that the functor of morphisms (of schemes) $\mathbf{Hom}(G,H)$ 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 algebraic and the unipotent radical of $G_{\bar{k}}$ is trivial, we show that the functor of group homomorphisms $\mathbf{Hom}_{\rm{gp}}(G,H)$ is represented by a sum of schemes of finite type. Along the way, we obtain rigidity properties for morphisms in the above settings.