Descent and generation for noncommutative coherent algebras over schemes

TL;DR

This paper explores descent properties of the bounded derived category of noncommutative coherent algebras over schemes, revealing new examples of strong generators and extending classical results into the noncommutative realm.

Contribution

It establishes descent results for strong generation in derived categories of noncommutative algebras over schemes, including new insights for Azumaya algebras.

Findings

Descent of strong generation in derived categories over various topologies.

New examples of schemes with strongly generated derived categories.

Applications to Azumaya algebras and noncommutative geometry.

Abstract

Our work shows forms of descent, in the fppf, h and \'{e}tale topologies, for strong generation of the bounded derived category of a noncommutative coherent algebra over a scheme. Even for (commutative) schemes this yields new perspectives. As a consequence we exhibit new examples where these bounded derived categories admit strong generators. We achieve our main results by leveraging the action of the scheme on the coherent algebra, allowing us to lift statements into the noncommutative setting. In particular, this leads to interesting applications regarding generation for Azumaya algebras.