  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.
