Deformations of categories of coherent sheaves via quivers with relations

TL;DR

This paper provides a combinatorial framework to understand deformations of categories of coherent sheaves on schemes using quivers with relations, enabling explicit analysis and applications to noncommutative singularity deformations.

Contribution

It introduces a combinatorial description of deformation theory for coherent sheaves via quivers, including criteria for algebraization and concrete examples.

Findings

Explicit combinatorial description of deformations

Criteria for algebraization of formal deformations

Applications to noncommutative singularity deformations

Abstract

We give an explicit combinatorial description of the deformation theory of the Abelian category of (quasi)coherent sheaves on any separated Noetherian scheme $X$ via the deformation theory of path algebras of quivers with relations, by using any affine open cover of $X$, or any tilting bundle on $X$, if available. We also give sufficient criteria for obtaining algebraizations of formal deformations, in which case the deformation parameters can be evaluated to a constant and the deformations can be compared to the original Abelian category on equal terms. We give concrete examples as well as applications to the study of noncommutative deformations of singularities.