We use the framework of perfectoid big Cohen-Macaulay (BCM) algebras to define a class of singularities for pairs in mixed characteristic, which we call purely BCM-regular singularities, and a corresponding adjoint ideal. We prove that these satisfy adjunction and inversion of adjunction with respect to the notion of BCM-regularity and the BCM test ideal defined by the first two authors. We compare them with the existing equal characteristic purely log terminal (PLT) and purely Fregular singularities and adjoint ideals. As an application, we obtain a uniform version of the Briancon-Skoda theorem in mixed characteristic. We also use our theory to prove that two-dimensional Kawamata log terminal singularities are BCM-regular if the residue characteristic p > 5, which implies an inversion of adjunction for three-dimensional PLT pairs of residue characteristic p > 5. In particular, divisorial centers of PLT pairs in dimension three are normal when p > 5. Furthermore, in Appendix A we provide a streamlined construction of perfectoid big Cohen-Macaulay algebras and show new functoriality properties for them using the perfectoidization functor of Bhatt and Scholze.
All Science Journal Classification (ASJC) codes
- Algebra and Number Theory
- Geometry and Topology