Compact approximation and descent for algebraic stacks

TL;DR

The paper addresses how to approximate objects in the derived category $D_{\operatorname{qc}}(\mathcal{X})$ of a Noetherian algebraic stack by compact complexes, and how such generation behaves under base change and descent. It develops a robust framework of approximation by compact complexes, proves a descent theorem along quasi-finite flat covers via dévissage, and then applies tensor-triangular methods to connect generation across morphisms. Key contributions include extending Lipman-Neeman type results to stacks under quasi-finite/diagonal or good moduli hypotheses, establishing an étale descent for approximation, and deriving base-change and Rouquier-dimension bounds for stacks (notably tame stacks with coarse moduli spaces). These results yield practical tools for understanding when $D^{b}_{\operatorname{coh}}(\mathcal{X})$ and $D_{\operatorname{qc}}(\mathcal{X})$ are generated by compact objects, with implications for moduli spaces and coarse moduli theory through generation and descent.

Abstract

This work focuses on approximation and generation for the derived category of complexes with quasi-coherent cohomology on algebraic stacks. Our methods establish that approximation by compact objects descends along covers that are quasi-finite and flat. This generalizes a result of Lipman--Neeman for schemes and extends a related result known for algebraic spaces. We also study the behavior of generation under the derived pushforward and pullback of a morphism between algebraic stacks.

Theorems & Definitions (65)

• Theorem A
• Theorem B
• Theorem C
• Theorem D
• Theorem E
• Remark 1.1
• Example 2.2
• Lemma 2.3
• proof
• Remark 2.4