Purity for flat cohomology
| Autor: | Cesnavicius, Kestutis, Scholze, Peter |
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| Rok vydání: | 2019 |
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| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: 97 pages; final version, to appear in Annals of Mathematics |
| Databáze: | arXiv |
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