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COUNTING GEOMETRIC BRANCHES VIA THE FROBENIUS MAP AND F-NILPOTENT SINGULARITIES

Part of: Ring extensions and related topics Commutative algebra: Homological methods General commutative ring theory

Published online by Cambridge University Press:  27 February 2024

HAILONG DAO Open the ORCID record for HAILONG DAO [Opens in a new window] ,
KYLE MADDOX Open the ORCID record for KYLE MADDOX [Opens in a new window]  and
VAIBHAV PANDEY Open the ORCID record for VAIBHAV PANDEY [Opens in a new window]
HAILONG DAO
Affiliation:
Department of Mathematics University of Kansas 1460 Jayhawk Boulevard Lawrence, Kansas 66045 United States hdao@ku.edu
KYLE MADDOX*
Affiliation:
Department of Mathematical Sciences University of Arkansas 850 West Dickson Street Fayetteville, Arkansas 72701 United States
VAIBHAV PANDEY
Affiliation:
Department of Mathematics Purdue University 150 North University Street West Lafayette, Indiana 47907 United States pandey94@purdue.edu
*
kmaddox@uark.edu
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Abstract

We give an explicit formula to count the number of geometric branches of a curve in positive characteristic using the theory of tight closure. This formula readily shows that the property of having a single geometric branch characterizes F-nilpotent curves. Further, we show that a reduced, local F-nilpotent ring has a single geometric branch; in particular, it is a domain. Finally, we study inequalities of Frobenius test exponents along purely inseparable ring extensions with applications to F-nilpotent affine semigroup rings.

Keywords

F-nilpotent ringsgeometric branchesintegral closureweak normalization

MSC classification

Primary: 13A35: Characteristic $p$ methods (Frobenius endomorphism) and reduction to characteristic $p$; tight closure
Secondary: 13D45: Local cohomology 13B40: Étale and flat extensions; Henselization; Artin approximation

Information

Type
Article
Information
Nagoya Mathematical Journal , Volume 255 , September 2024 , pp. 724 - 741
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Foundation Nagoya Mathematical Journal

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