Hom schemes for algebraic groups

TL;DR

This work extends the representability of Hom schemes from reductive to non-reductive group schemes G over a base S, proving that under mild hypotheses the functor $rac{Hom}_{S ext{-gp}}(G,H)$ is representable (as a disjoint union of schemes, often affine or quasi-affine) and clarifying when base change preserves this structure. A core innovation is a general representability criterion built around purity and a notion of strong generation by finitely many subtori, allowing reduction to Hom data on tori and a finite collection of subschemes. The paper then derives field-specific results (characterizing closed $H$-orbits in Hom in terms of Serre’s complete reducibility, among other statements), analyzes Isom schemes, and develops characteristic-0 refinements, while also addressing global-base phenomena, generic vs. global representability, and functorial behavior under restriction to tori or parabolics. The results yield a flexible framework for representability across bases and provide numerous examples, including parabolic cases, Moy–Prasad-type filtrations, and non-reductive contexts, with implications for orbit structure and moduli in algebraic group theory.

Abstract

In SGA3, Demazure and Grothendieck showed that if $G$ and $H$ are smooth affine group schemes over a scheme $S$ and $G$ is reductive, then the functor of $S$-homomorphism $G \to H$ is representable. In this paper we extend this result to cover cases in which $G$ is not reductive, with much simpler proofs. Our results apply in particular to parabolics over any base, and they are essentially optimal over a field. We also relate the closed orbits in Hom schemes to Serre's theory of complete reducibility, answer a question of Furter--Kraft, and provide many examples.

Theorems & Definitions (110)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Definition 2.3
• Lemma 2.4
• proof