Vol. 12, No. 1, 2018

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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
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