Česnavičius, Kęstutis; Scholze, Peter Purity for flat cohomology. (English) Zbl 1528.14024 Ann. Math. (2) 199, No. 1, 51-180 (2024). In this paper, the authors prove purity theorem for flat cohomology. The following main theorem is proved: For a Noetherian local ring \((R,\mathfrak{m})\) that is a complete intersection (this means that its completion is a quotient of a regular ring by a regular sequence) and a commutative, finite flat \(R\)-group scheme \(G\), \[ H^i_{\mathfrak{m}}(R,G)\simeq 0 \quad \text{for} \begin{cases} i< \dim R, & \\ i \leq \dim R &\text{ if }R\text{ is regular and not a field}. \end{cases} \] This is a flat cohomology version of the Gabber-Thomason purity theorem for étale cohomology that has been conjectured by Grothendieck. Gabber proved the following: For a regular local ring \((R,\mathfrak{m})\) and a commutative, finite étale \(R\)-group \(G\) whose order is invertible in \(R\), \[ H_{\mathfrak{m}}^i(R,G)=0 \quad \text{for }i<2 \dim R. \] Two proofs of this fact have been known. They give a third proof of this theorem using perfectoid techniques to reduce to the positive characteristic case.As a corollary of the main theorem, they settle two conjectures of Gabber: Let \((R,\mathfrak{m})\) be a Noetherian local ring that is a complete intersection and let \(U_R:=\mathrm{Spec}\ R \setminus \{\mathfrak{m}\}\). (a) If \(\dim R \geq 3\), \(\mathrm{Pic}(U_R)_{\mathrm{tors}}\simeq 0\).(b) If \(\dim R \geq 4\) or if both \(R\) is regular and \(\dim R \geq 2\), then \[ \mathrm{Br}(R) \xrightarrow{\sim} \mathrm{Br}(U_R). \] Furthermore, they prove the global version of (b): For a Noetherian scheme \(X\) and a closed subset \(Z\subset X\) such that each \(\mathcal{O}_{X,z}\) with \(z \in Z\) is either a complete intersection of dimension \(\geq 4\) or regular of dimension \(\geq 2\), then \[ H^2(X, \mathbb{G}_{\mathrm m})_{\mathrm{tors}} \xrightarrow{\sim} H (X \setminus Z, \mathbb{G}_{\mathrm m})_{\mathrm{tors}}. \] The main theorem is reduced to the purity theorem for flat cohomology in a perfectoid setting in the following. Let \(p\) be a prime number. For a perfectoid \(\mathbb{Z}_p\)-algebra \(A\), a commutative, finite, locally free \(A\)-group \(G\) of \(p\)-power order, and a closed subset \(Z\subset \mathrm{Spec}\ A/pA\) such that \(\mathrm{depth}_ Z(A) \geq d\) in the sense that there is an \(A\)-regular sequence \(a_1, \ldots, a_d \in A\) that vanishes on \(Z\), we have \[ H_Z^i(A,G) \simeq 0 \quad \text{for }i<d. \] To show the main theorem and this theorem, they 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. The key formula is \[ R\Gamma_Z(A,G)\simeq R\Gamma_Z(\mathbb{A}_{\mathrm{Inf}}(A),\mathbb{M}(G))^{V-1}, \] where \(\mathbb{M}(G)\) denotes the crystalline Dieudonné module. Reviewer: Takahiro Tsushima (Chiba) Cited in 14 Documents MSC: 14F20 Étale and other Grothendieck topologies and (co)homologies 14F22 Brauer groups of schemes 14F30 \(p\)-adic cohomology, crystalline cohomology 14H20 Singularities of curves, local rings Keywords:Brauer group; complete intersection; purity; perfectoid; flat cohomology; animated ring × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] Grothendieck, Alexander, Rev\^etements \'etales et groupe fondamental (SGA 1), Documents Math\'ematiques (Paris), 3, xviii+327 pp. (2003) [2] Grothendieck, Alexander; {A. Laszlo (ed.)}, Cohomologie Locale des Faisceaux Coh\'{e}rents et Th\'{e}or\`emes de {L}efschetz Locaux et Globaux ({SGA} 2), Doc. Math. (Paris), 4, x+208 pp. (2005) · Zbl 1079.14001 [3] Gille, Philippe; Polo, Patrick, Sch\'emas en groupes {(SGA 3)}. {T}ome {III. S}tructure des Sch\'emas en Groupes R\'eductifs, Doc. Math. (Paris), 8, lvi+337 pp. (2011) [4] Artin, M.; Grothendieck, A.; Verdier, J. 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