Purity for the Brauer group

Abstract

A purity conjecture due to Grothendieck and Auslander--Goldman predicts that the Brauer group of a regular scheme does not change after removing a closed subscheme of codimension $\ge 2$. The combination of several works of Gabber settles the conjecture except for some cases that concern $p$-torsion Brauer classes in mixed characteristic $(0,p)$. We establish the remaining cases by using the tilting equivalence for perfectoid rings. To reduce to perfectoids, we control the change of the Brauer group of the punctured spectrum of a local ring when passing to a finite flat cover.

17 pages; final version, to appear in Duke Mathematical Journal

Cited by in corpus (9)

• The Brauer group of the moduli stack of elliptic curves
• A variant of perfectoid Abhyankar's lemma and almost Cohen-Macaulay algebras
• Evaluating the wild Brauer group
• Perfectoid towers and their tilts : with an application to the étale cohomology groups of local log-regular rings
• Unramified Cohomology of Quadrics in Characteristic Two
• The cohomological Brauer group of weighted projective spaces and stacks
• Degree 2 cohomological invariants of linear algebraic groups
• On the Brauer groups of fibrations
• Vanishing of Brauer groups of moduli stacks of stable curves