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 for $i \le \mathrm{dim}(R)$. 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.