Purity for flat cohomology

Abstract

We establish the flat cohomology version of the Gabber--Thomason purity for étale cohomology: for a complete intersection Noetherian local ring $(R, \mathfrak{m})$ and a commutative, finite, flat $R$-group $G$, the flat cohomology $H^i_\mathfrak{m}(R, G)$ vanishes for $i \lt \mathrm{dim}(R)$. For small $i$, this settles conjectures of Gabber that extend the Grothendieck--Lefschetz theorem and give purity for the Brauer group for schemes with complete intersection singularities. For the proof, we reduce to a flat purity statement for perfectoid rings, establish $p$-complete arc descent for flat cohomology of perfectoids, and then relate to coherent cohomology of $\mathbb{A}_{\mathrm{Inf}}$ via prismatic Dieudonné; theory. We also present an algebraic version of tilting for étale cohomology, use it to reprove the Gabber--Thomason purity, and exhibit general properties of fppf cohomology of (animated) rings with finite, locally free group scheme coefficients, such as excision, agreement with fpqc cohomology, and continuity.