Coherently complete algebraic stacks in positive characteristic

TL;DR

This paper investigates the coherent completeness of algebraic stacks in positive characteristic, proposing a conjecture for quotient stacks and proving it in specific cases, with implications for local structure theorems.

Contribution

The paper introduces a conjecture on coherent completeness of quotient stacks in positive characteristic and proves it in cases where the invariant ring is artinian or the group acts trivially.

Findings

Conjecture on coherent completeness of quotient stacks in positive characteristic.

Proof of the conjecture when the invariant ring is artinian.

Proof of the conjecture when the group acts trivially on the spectrum.

Abstract

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.