Centers of perfectoid purity

We introduce a mixed characteristic analog of log canonical centers in characteristic <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>0</mml:mn> </mml:math> and centers of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>F</mml:mi> </mml:math> -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.