Purity for the Brauer group
The purity conjecture is a famous conjecture due to \textit{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 [\textit{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 \[ H^3_{\text{et}} \left(X, \mathbb{G}_m\right)\hookrightarrow H^3_{\text{et}} \left(X - Z, \mathbb{G}_m\right). \]
scientific article