Homomorphisms of algebraic groups: representability and rigidity

2021/01/29 by Michel Brion, Brion, Michel
Chemistry · Mathematics · #14D20 #14K05 #14L15 (Primary) #14L30 #20G15 (Secundary) #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Axial and Atropisomeric Chirality Synthesis #FOS: Mathematics

paper · doi:10.48550/arxiv.2101.12460

openalex publication_date 2021/01/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given two algebraic groups G, H over a field k, we investigate the representability of the functor of morphisms (of schemes) Hom(G,H) and the subfunctor of homomorphisms (of algebraic groups) Hom_{\rm gp}(G,H). We show that Hom(G,H) is represented by a group scheme, locally of finite type, if the k-vector space O(G) is finite-dimensional; the converse holds if H is not étale. When G is linearly reductive and H is smooth, we show that Hom_{\rm gp}(G,H) is represented by a smooth scheme M; moreover, every orbit of H acting by conjugation on M is open.

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