Descending strong generation in algebraic geometry
Timothy De Deyn, Pat Lank, Kabeer Manali Rahul
TL;DR
The paper develops a presheaf-of-triangulated-categories framework to study Zariski descent of strong generation for algebro-geometric triangulated categories, with Mayer–Vietoris triangles and boundedness conditions providing the technical core. It proves a general descent theorem for strong generation along Zariski squares, yielding that $D_{\operatorname{sg}}(X)$, $D^b_{\operatorname{coh}}(X)$, and $\operatorname{Perf}(X)$ inherit strong generators from local data on a cover, and recovers known results (Neeman 2021, Lank 2024) while extending to algebraic stacks via a quasi-DM descent theorem. The work also develops big-generator variants and discusses descent beyond Zariski, including stack applications, using descendable objects and étale-type covers. Overall, it provides a unified, flexible criterion to verify strong generation locally and extend to global contexts in schemes and stacks, with broad implications for derived-category methods in algebraic geometry.
Abstract
We formalize the main approach for showing Zariski descent-type statements for strong generation of triangulated categories associated to algebro-geometric objects. This recovers various known statements in the literature. As applications we show that strong generation for the singularity category of a Noetherian separated scheme is Zariski local and obtain a strong generation result for the bounded derived category of a Noetherian concentrated algebraic stacks with finite diagonal.