Appears in collection : Rational points on irrational varieties

For regular Noetherian schemes, the cohomological Brauer group is insensitive to removing a closed subscheme of codimension ≥ 2. I will discuss the corresponding statement for schemes with local complete intersection singularities, for instance, for complete intersections in projective space. Such purity phenomena turn out to be low cohomological degree cases of purity for flat cohomology. I will discuss the latter from the point of view of the perfectoid approach to such questions. The talk is based on joint work with Peter Scholze.