Purity for flat cohomology

By Kęstutis Česnavičius, Peter Scholze. Annals of Mathematics, 2019

81 citations recorded by the source corpus.

We establish the flat cohomology version of the Gabber-Thomason purity for etale 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 Dieudonne theory. We also present an algebraic version of tilting for etale cohomology, use it 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 $p$-adic continuity.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f3e2eb8bad033ecdb433a59e1c6e7b2a7be9ec35