Moduli Problems in Abelian Categories and the Reconstruction Theorem

TL;DR

This paper provides a moduli-theoretic approach to reconstruct schemes from their categories of quasi-coherent sheaves, extending classical results to algebraic spaces and analyzing autoequivalences and gerbes.

Contribution

It introduces elementary methods for reconstructing schemes from sheaf categories, extends theorems to algebraic spaces, and explores autoequivalences and gerbes beyond classical cases.

Findings

Reconstruction of schemes from quasi-coherent sheaves via moduli theory

Extension of Gabriel's theorem to algebraic spaces

Identification of autoequivalences with automorphisms and line bundle twists

Abstract

We give a moduli-theoretic proof of the classical theorem of Gabriel, stating that a scheme can be reconstructed from the abelian category of quasi-coherent sheaves over it. The methods employed are elementary and allow us to extend the theorem to (quasi-compact and separated) algebraic spaces. Using more advanced technology (and assuming flatness) we also give a proof of the folklore result that the group of autoequivalences of the category of quasi-coherent sheaves consists of automorphisms of the underlying space and twists by line bundles. We apply our strategy to prove analogous statements for categories of sheaves twisted by a Gm-gerbe. Our methods allow us to treat even gerbes not coming from a Brauer class. As a pleasant consequence, we deduce a Morita theory for sheaves of abelian categories.