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

Polstra, Thomas; Tucker, Kevin

doi:10.2140/ant.2018.12.61

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

Thomas Polstra and Kevin Tucker

Vol. 12 (2018), No. 1, 61–97

DOI: 10.2140/ant.2018.12.61

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.

Keywords

$F$-signature, Hilbert–Kunz multiplicity

Mathematical Subject Classification 2010

Primary: 13A35

Secondary: 14B05

Milestones

Received: 23 January 2017

Revised: 18 September 2017

Accepted: 31 October 2017

Published: 13 March 2018

Authors

Thomas Polstra
Department of Mathematics University of Utah Salt Lake City, UT United States

Kevin Tucker
Department of Mathematics University of Illinois at Chicago Chicago, IL United States

Vol. 12, No. 1, 2018