Further remarks on derived categories of algebraic stacks

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.

24 pages, comments welcome!