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

TL;DR

This paper demonstrates that two fundamental properties of derived categories for schemes do not generally hold for algebraic stacks in positive characteristic, revealing limitations in extending scheme results to stacks.

Contribution

It provides the first known counterexamples showing the failure of these properties for algebraic stacks in positive characteristic.

Findings

The unbounded derived category of algebraic stacks in positive characteristic is not always compactly generated by perfect complexes.

The functor from the derived category of quasi-coherent sheaves to the unbounded derived category of the stack is not always an equivalence.

These results highlight fundamental differences between schemes and algebraic stacks in positive characteristic.

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 $\Psi_X \colon \mathsf{D}(\mathsf{QCoh}(X)) \to \mathsf{D}_{\mathrm{qc}}(X)$ is an equivalence. Our main results are that for algebraic stacks in positive characteristic, the assertions (1) and (2) are typically false.