Perfectoid signature, perfectoid HilbertKunz multiplicity, and an application to local fundamental groups
Abstract
We define a (perfectoid) mixed characteristic version of $F$signature and HilbertKunz multiplicity by utilizing the perfectoidization functor of BhattScholze and Faltings' normalized length (also developed in the work of GabberRamero). 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 HilbertKunz multiplicity is 1 and we show that perfectoid HilbertKunz multiplicity characterizes BCM closure and extended plus closure of $\mathfrak{m}$primary ideals. We demonstrate that perfectoid signature detects BCMregularity and transforms similarly to $F$signature or normalized volume under quasiétale maps. As a consequence, we prove that BCMregular 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 BCMrational singularities.
 Publication:

arXiv eprints
 Pub Date:
 September 2022
 DOI:
 10.48550/arXiv.2209.04046
 arXiv:
 arXiv:2209.04046
 Bibcode:
 2022arXiv220904046C
 Keywords:

 Mathematics  Commutative Algebra;
 Mathematics  Algebraic Geometry;
 Mathematics  Number Theory;
 14G45;
 13A35;
 14F35;
 14B05;
 13D22;
 14F18;
 13C20;
 14C20;
 14D10;
 11G99
 EPrint:
 75 pages, applications and examples added, other minor changes, comments welcome