SINGULARITIES IN MIXED CHARACTERISTIC VIA PERFECTOID BIG COHEN-MACAULAY ALGEBRAS

We utilize recent results of Andre and Gabber on the existence of weakly functorial, integral perfectoid big Cohen-Macaulay (BCM) algebras to study singularities of local rings in mixed characteristic. In particular, we introduce a mixed characteristic BCM-variant of rational/ F -rational singularities, of log terminal/ F-regular singularities, and of multiplier/test ideals of divisor pairs. We prove a number of results about these objects including a restriction theorem for perfectoid BCM multiplier/test ideals and deformation statements for perfectoid BCM-regular and BCM-rational singularities. As an application, we obtain results on the behavior of F-regular and F-rational singularities in arithmetic families.