Perfectoid signature, perfectoid Hilbert-Kunz multiplicity, and an application to local fundamental groups

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.