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

Understand

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.