Purity for the Brauer group. (English) Zbl 1430.14047

The purity conjecture is a famous conjecture due to A. Grothendieck [Adv. Stud. Pure Math. 3, 88–188 (1968; Zbl 0198.25901)]. It predicts that the Brauer group of a regular scheme \(X\) is not changed when removing a closed subscheme \(Z \subset X\) of codimension at least two. Combining several results due to Grothendieck and Gabber, the conjecture was proved except in some cases. In the present paper, the author settles the these remaining cases. They concern \(p\)-torsion Brauer classes in mixed characteristic \((0,p)\). This finishes the proof of the following theorem.
Theorem [K. Česnavičius, Duke Math. J. 168, No. 8, 1461–1486 (2019; Zbl 07080116)]. For each locally noetherian scheme \(X\) and a closed subscheme \(Z\subset X\), such that for every \(z \in Z\) the local ring \(\mathcal{O}_{X,z}\) of \(X\) at \(z\) is regular of dimension \(\geq 2\), we have \[ H^2_{\text{et}} \left(X, \mathbb{G}_m\right) \overset{\sim}{\longrightarrow} H^2_{\text{et}} \left(X - Z, \mathbb{G}_m\right) \] and \[ H^3_{\text{et}} \left(X, \mathbb{G}_m\right)\hookrightarrow H^3_{\text{et}} \left(X - Z, \mathbb{G}_m\right). \]


14F22 Brauer groups of schemes
14F20 Étale and other Grothendieck topologies and (co)homologies
14G22 Rigid analytic geometry
16K50 Brauer groups (algebraic aspects)


Zbl 0198.25901
Full Text: DOI arXiv Euclid


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