Purity for flat cohomology | Annals of Mathematics

IF 8.3 2区 材料科学 Q1 MATERIALS SCIENCE, MULTIDISCIPLINARY
Kęstutis Česnavičius, Peter Scholze
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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.

平面同调的纯度 | 数学年鉴
我们建立了伽伯-托马森纯度的平同调版本:对于完全交点诺特局部环 $(R, \mathfrak {m})$和交换、有限、平 $R$- 群 $G$,平同调 $H^i_\mathfrak {m}(R,G)$在 $i \le \mathrm{dim}(R)$ 时消失。对于较小的 $i$,这解决了加伯尔的猜想,即扩展格罗内迪克-勒夫谢茨定理,并给出具有完全交点奇点的方案的布劳尔群的纯度。为了证明这一点,我们还原了完形环的平面纯度声明,为完形的平面同调建立了 $p$ 完整的弧降,然后通过棱柱迪厄多内理论将其与 $\mathbb {A}_{\mathrm {Inf}}$ 的相干同调联系起来。我们还提出了一个倾斜的代数版本的 étale cohomology,用它来重新证明 Gabber-Thomason 纯度,并展示了具有有限局部自由群方案系数的(动画)环的 fppf cohomology 的一般性质,如切除、与 fpqc cohomology 一致和连续性。
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来源期刊
ACS Applied Materials & Interfaces
ACS Applied Materials & Interfaces 工程技术-材料科学:综合
CiteScore
16.00
自引率
6.30%
发文量
4978
审稿时长
1.8 months
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.
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