An analog of adjoint ideals and PLT singularities in mixed characteristic

We use the framework of perfectoid big Cohen-Macaulay 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 PLT and purely $F$-regular singularities and adjoint ideals. As an application, we obtain a uniform version of the Brian\c{c}on-Skoda theorem in mixed characteristic. We also use our theory to prove that two-dimensional KLT 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 the appendix 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.