Comparison theorems for deformation functors via invariant theory

TL;DR

This paper explores how deformations of algebraic schemes relate to invariant theory, extending classical comparison theorems to broader contexts involving quotients by reductive groups.

Contribution

It generalizes comparison theorems for deformations from projective schemes to schemes arising as quotients by linearly reductive groups, linking algebraic and invariant deformations.

Findings

Conditions for smooth comparison morphisms

Criteria for isomorphisms in deformation comparisons

Extension of classical theorems to quotient schemes

Abstract

We compare deformations of algebras to deformations of schemes in the setting of invariant theory. Our results generalize comparison theorems of Schlessinger and the second author for projective schemes. We consider deformations (abstract and embedded) of a scheme $X$ which is a good quotient of a quasi-affine scheme $X^\prime$ by a linearly reductive group $G$ and compare them to invariant deformations of an affine $G$-scheme containing $X^\prime$ as an open invariant subset. The main theorems give conditions for when the comparison morphisms are smooth or isomorphisms.