Further remarks on derived categories of algebraic stacks

TL;DR

This paper extends derived category equivalences for algebraic stacks with quasi-affine diagonals over characteristic zero fields and proves compact generation of the unbounded derived category in the smooth case.

Contribution

It generalizes known equivalences to unbounded categories and establishes compact generation for smooth stacks, using descendable algebras and extending results to positive and mixed characteristics.

Findings

Extended derived category equivalence to unbounded categories.

Proved compact generation of the derived category for smooth stacks.

Established results in positive and mixed characteristics.

Abstract

Let $X$ be an algebraic stack with quasi-affine diagonal of finite type over a field $k$ of characteristic $0$. We extend the well-known equivalence $\mathsf{D}^+(\mathsf{QCoh}(X)) \simeq \mathsf{D}_{\mathrm{qc}}^+(X)$ to unbounded derived categories. We also prove that if $X$ is smooth over $k$, then $\mathsf{D}_{\mathrm{qc}}(X)$ is compactly generated. We accomplish the former using the descendable algebras of Mathew. We also establish related results in positive and mixed characteristics.