Algebra & Number Theory

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

Thomas Polstra and Kevin Tucker

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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]

$F$-signature Hilbert–Kunz multiplicity


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.

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