A reconstruction theorem for abelian categories of twisted sheaves

TL;DR

This paper proves a reconstruction theorem for abelian categories of twisted sheaves on schemes, confirming Cldraru's conjecture by leveraging Rosenberg's ideas and To"en's work on derived Azumaya algebras.

Contribution

It introduces a new reconstruction theorem for abelian categories of twisted sheaves, extending existing frameworks and confirming a longstanding conjecture.

Findings

Proved a reconstruction theorem for twisted sheaves on schemes.

Confirmed C0ld0raru's conjecture using derived Azumaya algebras.

Extended the understanding of abelian categories in algebraic geometry.

Abstract

We use an idea of Rosenberg to prove a reconstruction theorem for abelian categories of alpha-twisted quasi-coherent sheaves on quasi-compact and quasi-separated schemes X when alpha is in the Brauer group of X. By applying the work of To\"en on derived Azumaya algebras, we give a proof of C\u{a}ld\u{a}raru's conjecture.