Purity for flat cohomology

TL;DR

This paper proves a flat cohomology version of the Gabber-Thomason purity theorem for complete intersection local rings, extending classical purity results and establishing new connections with perfectoid rings, prismatic theory, and tilting.

Contribution

It establishes flat cohomology purity for complete intersection rings, introduces a flat purity statement for perfectoid rings, and develops an algebraic tilting approach for tale cohomology.

Findings

Vanishing of flat cohomology for certain degrees in complete intersection rings.

Reduction of purity to perfectoid rings and establishment of arc descent.

Reproof of Gabber-Thomason purity and properties of fppf cohomology.

Abstract

We establish the flat cohomology version of the Gabber-Thomason purity for \'{e}tale 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 $i < \mathrm{dim}(R)$. 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 Dieudonn\'{e} theory. We also present an algebraic version of tilting for \'{e}tale 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.