{"title":"Purity for flat cohomology | Annals of Mathematics","authors":"Kęstutis Česnavičius, Peter Scholze","doi":"10.4007/annals.2024.199.1.2","DOIUrl":null,"url":null,"abstract":"<p>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.</p>\n<p></p>","PeriodicalId":5,"journal":{"name":"ACS Applied Materials & Interfaces","volume":null,"pages":null},"PeriodicalIF":8.3000,"publicationDate":"2023-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"ACS Applied Materials & Interfaces","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4007/annals.2024.199.1.2","RegionNum":2,"RegionCategory":"材料科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATERIALS SCIENCE, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 0
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.
期刊介绍:
ACS Applied Materials & Interfaces is a leading interdisciplinary journal that brings together chemists, engineers, physicists, and biologists to explore the development and utilization of newly-discovered materials and interfacial processes for specific applications. Our journal has experienced remarkable growth since its establishment in 2009, both in terms of the number of articles published and the impact of the research showcased. We are proud to foster a truly global community, with the majority of published articles originating from outside the United States, reflecting the rapid growth of applied research worldwide.