One positive and two negative results for derived categories of algebraic stacks

Abstract

Let $X$ be a quasi-compact and quasi-separated scheme. There are two fundamental and pervasive facts about the unbounded derived category of $X$: (1) ${\mathsf{D}}_{\mathrm{qc}}(X)$ is compactly generated by perfect complexes and (2) if $X$ is noetherian or has affine diagonal, then the functor ${\Psi }_X:\mathsf{D}(\mathsf{QCoh}(X))\to {\mathsf{D}}_{\mathrm{qc}}(X)$ is an equivalence. Our main results are that for algebraic stacks in positive characteristic, the assertions (1) and (2) are typically false.

arXiv commentmain results strengthened in the noetherian situation; rewrite of Introduction; additional details on well-generation