Purity phenomena in arithmetic geometry show that in sufficiently nonsingular situations finite étale covers or cohomology classes extend uniquely over closed subschemes of large codimension. I will overview this in the case of the Brauer group and, more generally, in the case of fppf cohomology with finite flat group scheme coefficients. I will stress situations in which the order of the cohomology classes is not invertible on the base scheme, where a recent method is to use perfectoid techniques to reduce such questions to equal characteristic.