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

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) ${\mathsf{D}}_{\mathrm{q}\mathrm{c}}(X)$ is compactly generated by perfect complexes and (2) if $X$ is noetherian or has affine diagonal, then the functor $\Psi_X \colon $\mathsf{D}$($\mathsf{Q}\mathsf{C}\mathsf{o}\mathsf{h}$(X)) \to $\mathsf{D}$_{$\mathrm{q}\mathrm{c}$}(X)$ is an equivalence. Our main results are that for algebraic stacks in positive characteristic, the assertions (1) and (2) are typically false.