We introduce a mixed characteristic analog of log canonical centers in characteristic 0 and centers of F-purity in positive characteristic, which we call centers of perfectoid purity. We show that their existence detects (the failure of) normality of the ring. We also show the existence of a special center of perfectoid purity that detects the perfectoid purity of the ring, analogously to the splitting prime of Aberbach and Enescu, and investigate its behavior under étale morphisms.
Anne Fayolle (Mon,) studied this question.