Algebroid prestacks and deformations of ringed spaces

TL;DR

This paper explores how deformations of categories of sheaves on ringed spaces can be described using algebroid prestacks, establishing a deformation equivalence between these categories for certain schemes.

Contribution

It demonstrates that deformations of sheaf categories are captured by algebroid prestacks, extending the framework to quasi-coherent sheaves on quasi-compact separated schemes.

Findings

Deformations of Mod(O) are obtained from algebroid prestacks.

For quasi-compact separated schemes, Qch(O) deformations are also described by algebroid prestacks.

There is a deformation equivalence between Mod(O) and Qch(O).

Abstract

For a ringed space (X,O), we show that the deformations of the abelian category Mod(O) of sheaves of O-modules are obtained from algebroid prestacks, as introduced by Kontsevich. In case X is a quasi-compact separated scheme the same is true for Qch(O), the category of quasi-coherent sheaves on X. It follows in particular that there is a deformation equivalence between Mod(O) and Qch(O).