  With the long-term goal of proving local structure theorems of algebraic
stacks in positive characteristic near points with reductive (but possibly
non-linearly reductive) stabilizer, we conjecture that quotient stacks of the
form $[\mathrm{Spec}\, A/G]$, with $G$ reductive and $A^G$ complete local, are
coherently complete along the unique closed point. We establish this conjecture
in two interesting cases: (1) $A^G$ is artinian and (2) $G$ acts trivially on
$\mathrm{Spec}\, A$. We also establish coherent completeness results for graded
unipotent group actions. In order to establish these results, we prove a number
of foundational statements concerning cohomological and completeness properties
of algebraic stacks -- including on how these properties ascend and descend
along morphisms.
