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 . The combination of several works of Gabber settles the conjecture except for some cases that concern -torsion Brauer classes in mixed characteristic . 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.
Kęstutis Česnavičius. "Purity for the Brauer group." Duke Math. J. 168 (8) 1461 - 1486, 1 June 2019. https://doi.org/10.1215/00127094-2018-0057