An analogue of adjoint ideals and PLT singularities in mixed characteristic
An analogue of adjoint ideals and PLT singularities in mixed characteristic
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混合特征中伴随理想和PLT奇点的类比
DOI：
10.1090/jag/797
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发表时间：
2022
影响因子：
1.8
通讯作者：
Witaszek, Jakub
中科院分区：
文献类型：
--
作者：
Ma, Linquan;Schwede, Karl;Tucker, Kevin;Waldron, Joe;Witaszek, Jakub
我们利用完美类大Cohen-Macaulay代数的框架，定义了一类混合特征对的奇异点，我们称之为纯bcm -正则奇异点，以及相应的伴随理想。根据前两位作者定义的BCM正则性概念和BCM检验理想，证明了它们满足附合和附合反转。我们将它们与已有的等特征PLT、纯正则奇异点和伴随理想作了比较。作为一个应用，我们得到了Brian\c{c}on-Skoda定理在混合特征中的统一版本。我们还利用我们的理论证明了二维KLT奇点在残差特征上是bcm规则的，这意味着三维PLT残差特征对的共轭性的反演。特别是，三维PLT对的分心是正常的。此外，在附录中，我们给出了完美类大Cohen-Macaulay代数的流线构造，并利用Bhatt和Scholze的完美化函子证明了它们的新功能性质。
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-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, which implies an inversion of adjunction for three-dimensional PLT pairs of residue characteristic. In particular, divisorial centers of PLT pairs in dimension three are normal when. 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.