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Title: An analogue of adjoint ideals and PLT singularities in mixed characteristic
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.  more » « less
Award ID(s):
1952366 1901672 2101800 1952522 1801849 2306854
PAR ID:
10326771
Author(s) / Creator(s):
; ; ; ;
Date Published:
Journal Name:
Journal of Algebraic Geometry
Volume:
31
Issue:
3
ISSN:
1056-3911
Page Range / eLocation ID:
497 to 559
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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