## Algebra & Number Theory

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

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

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

#### 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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