Descending strong generation in algebraic geometry

TL;DR

This paper formalizes a method for proving Zariski descent for strong generation in algebraic geometry, unifying known results and applying them to singularity categories and algebraic stacks.

Contribution

It introduces a formal approach for Zariski descent of strong generation, recovering known results and extending to new classes of algebraic objects.

Findings

Strong generation for singularity categories is Zariski local.

Established strong generation for derived categories of algebraic stacks.

Unified framework for descent statements 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.