Etale twists in noncommutative algebraic geometry and the twisted Brauer space

TL;DR

This paper introduces the twisted Brauer space to classify etale twists of derived categories and associative algebras, providing complete classifications for certain geometric objects and partial results for others.

Contribution

It develops a new method using the twisted Brauer space for classifying etale twists, extending previous work on Brauer groups to noncommutative and derived settings.

Findings

Complete classification for genus 0 curves, quadrics, and noncommutative projective spaces

Partial classification results for higher genus curves

Builds on recent work with David Gepner on Brauer groups of ring spectra

Abstract

This paper studies etale twists of derived categories of schemes and associative algebras. A general method, based on a new construction called the twisted Brauer space, is given for classifying etale twists, and a complete classification is carried out for genus 0 curves, quadrics, and noncommutative projective spaces. A partial classification is given for curves of higher genus. The techniques build upon my recent work with David Gepner on the Brauer groups of commutative ring spectra.