Purity for the Brauer group

TL;DR

This paper proves the purity conjecture for the Brauer group of regular schemes, especially addressing the challenging p-torsion cases in mixed characteristic using perfectoid techniques.

Contribution

It completes the proof of the purity conjecture for the Brauer group by employing tilting equivalence for perfectoid rings to handle remaining cases.

Findings

The Brauer group remains unchanged after removing codimension ≥ 2 subschemes.

The conjecture is settled for p-torsion classes in mixed characteristic.

Control of Brauer group changes under finite flat covers is established.

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.