Perfectoid signature, perfectoid Hilbert-Kunz multiplicity, and an application to local fundamental groups
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866908432499474432 |
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| author | Cai, Hanlin Lee, Seungsu Ma, Linquan Schwede, Karl Tucker, Kevin |
| author_facet | Cai, Hanlin Lee, Seungsu Ma, Linquan Schwede, Karl Tucker, Kevin |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_04046 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Perfectoid signature, perfectoid Hilbert-Kunz multiplicity, and an application to local fundamental groups Cai, Hanlin Lee, Seungsu Ma, Linquan Schwede, Karl Tucker, Kevin Commutative Algebra Algebraic Geometry Number Theory 14G45, 13A35, 14F35, 14B05, 13D22, 14F18, 13C20, 14C20, 14D10, 11G99 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. |
| title | Perfectoid signature, perfectoid Hilbert-Kunz multiplicity, and an application to local fundamental groups |
| topic | Commutative Algebra Algebraic Geometry Number Theory 14G45, 13A35, 14F35, 14B05, 13D22, 14F18, 13C20, 14C20, 14D10, 11G99 |
| url | https://arxiv.org/abs/2209.04046 |