Singularities in mixed characteristic via perfectoid big Cohen–Macaulay algebras

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