Descent and generation for noncommutative coherent algebras over schemes
Timothy De Deyn, Pat Lank, Kabeer Manali Rahul
TL;DR
The paper studies strong generation of derived categories $D^b_{coh}(\mathcal{A})$ for coherent noncommutative algebras over schemes, proving descent results across Zariski, fppf, $h$, and étale topologies by leveraging the action of the base scheme on $\mathcal{A}$. It introduces diagonal dimension and a nc projection formula to lift generation statements from the commutative world to noncommutative settings, yielding stalk-local-to-global principles and explicit Rouquier-dimension bounds, especially for Azumaya algebras. Key contributions include equivalences for the existence of strong $\oplus$-generators under various coverings, proper descent bounds, and Azumaya-centered generation results that relate to the center and potential splitting phenomena. The results broaden the toolbox for understanding when bounded derived categories of noncommutative algebras admit strong generators and provide concrete consequences for Azumaya algebras and derived-splinter contexts, with implications for both noncommutative and commutative geometry.
Abstract
Our work shows forms of descent, in the fppf, h and étale 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.