Algebra & Number Theory

$F$-signature and Hilbert–Kunz multiplicity: a combined approach and comparison

Thomas Polstra and Kevin Tucker

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Abstract

We present a unified approach to the study of F -signature, Hilbert–Kunz multiplicity, and related limits governed by Frobenius and Cartier linear actions in positive characteristic commutative algebra. We introduce general techniques that give vastly simplified proofs of existence, semicontinuity, and positivity. Furthermore, we give an affirmative answer to a question of Watanabe and Yoshida allowing the F -signature to be viewed as the infimum of relative differences in the Hilbert–Kunz multiplicities of the cofinite ideals in a local ring.

Article information

Source
Algebra Number Theory, Volume 12, Number 1 (2018), 61-97.

Dates
Received: 23 January 2017
Revised: 18 September 2017
Accepted: 31 October 2017
First available in Project Euclid: 4 April 2018

Permanent link to this document
https://projecteuclid.org/euclid.ant/1522807231

Digital Object Identifier
doi:10.2140/ant.2018.12.61

Mathematical Reviews number (MathSciNet)
MR3781433

Zentralblatt MATH identifier
06861736

Subjects
Primary: 13A35: Characteristic p methods (Frobenius endomorphism) and reduction to characteristic p; tight closure [See also 13B22]
Secondary: 14B05: Singularities [See also 14E15, 14H20, 14J17, 32Sxx, 58Kxx]

Keywords
$F$-signature Hilbert–Kunz multiplicity

Citation

Polstra, Thomas; Tucker, Kevin. $F$-signature and Hilbert–Kunz multiplicity: a combined approach and comparison. Algebra Number Theory 12 (2018), no. 1, 61--97. doi:10.2140/ant.2018.12.61. https://projecteuclid.org/euclid.ant/1522807231


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