Deformations and automorphisms: a framework for globalizing local tangent and obstruction spaces

TL;DR

This paper develops a comprehensive framework for understanding how local deformation data of a scheme relate to global deformation properties, extending existing theories to more general stack-like contexts.

Contribution

It generalizes Schlessinger's deformation theory by incorporating automorphisms and local-global relationships within a unified, stack-based framework.

Findings

Many known deformation results are proved in full generality.

The framework formalizes local behavior of deformation problems.

It extends Rim's work on homogeneous groupoids.

Abstract

Building on Schlessinger's work, we define a framework for studying geometric deformation problems which allows us to systematize the relationship between the local and global tangent and obstruction spaces of a deformation problem. Starting from Schlessinger's functors of Artin rings, we proceed in two steps: we replace functors to sets by categories fibered in groupoids, allowing us to keep track of automorphisms, and we work with deformation problems naturally associated to a scheme X, and which naturally localize on X, so that we can formalize the local behavior. The first step is already carried out by Rim in the context of his homogeneous groupoids, but we develop the theory substantially further. In this setting, many statements known for a range of specific deformation problems can be proved in full generality, under very general stack-like hypotheses.