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Abstract Let be a submonoid of a free Abelian group of finite rank. We show that if is a field of prime characteristic such that the monoid ‐algebra is , then is a finitely generated ‐algebra, or equivalently, that is a finitely generated monoid. Split‐‐regular rings are possibly non‐Noetherian or non‐‐finite rings that satisfy the defining property of strongly ‐regular rings from the theories of tight closure and ‐singularities. Our finite generation result provides evidence in favor of the conjecture that rings in function fields over have to be Noetherian. The key tool is Diophantine approximation from convex geometry. 

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 F F -regular singularities and adjoint ideals. As an application, we obtain a uniform version of the Briançon-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 p>5 , which implies an inversion of adjunction for three-dimensional PLT pairs of residue characteristic p > 5 p>5 . In particular, divisorial centers of PLT pairs in dimension three are normal when p > 5 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.