论文标题
PerfectOid签名，完美的Hilbert-Kunz多重性，并应用于本地基本组
Perfectoid signature, perfectoid Hilbert-Kunz multiplicity, and an application to local fundamental groups
论文作者
论文摘要
我们通过利用Bhatt-Scholze和Faltings的正常长度（在Gabber-ramero的工作中开发），通过利用Bhatt-Scholze和Faltings的Perfectoidization函数来定义$ f $ signature和Hilbert-Kunz多重性的（完美的）混合特性版本。我们表明，这些定义与同等特征$ p> 0 $的经典理论一致。我们证明，只有当它的完美体签名或完美的Hilbert-kunz多重性是1，并且我们证明Perfectoid Hilbert-Kunz多重性表征BCM闭合和扩展以及$ M $ $ $ $ $ $ - $ $ - $ - $ $ - 主要理想。我们证明，完美的签名检测到BCM的定型性，并在准图中类似于$ f $ - 签名或归一化体积。结果，我们证明了BCM规范环具有有限的本地étale基本组，并且在其除数阶级群体中也有限扭转。最后，我们还定义了相对理性签名的混合特征版本，并表明它的特征是BCM理性的概念。
We define a (perfectoid) mixed characteristic version of $F$-signature and Hilbert-Kunz multiplicity by utilizing the perfectoidization functor of Bhatt-Scholze and Faltings' normalized length (also developed in the work of Gabber-Ramero). We show that these definitions coincide with the classical theory in equal characteristic $p > 0$. We prove that a ring is regular if and only if either its perfectoid signature or perfectoid Hilbert-Kunz multiplicity is 1 and we show that perfectoid Hilbert-Kunz multiplicity characterizes BCM closure and extended plus closure of $m$-primary ideals. We demonstrate that perfectoid signature detects BCM-regularity and transforms similarly to $F$-signature or normalized volume under quasi-étale maps. As a consequence, we prove that BCM-regular rings have finite local étale fundamental group and also finite torsion part of their divisor class groups. Finally, we also define a mixed characteristic version of relative rational signature, and show it characterizes BCM-rational singularities.
