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

2016/08/31 by Thomas Polstra, Kevin Tucker · 1 citation
Computer Science · Mathematics · #Advanced Combinatorial Mathematics #Algebra over a field #Commutative Algebra and Its Applications #Commutative property #Infimum and supremum #Mathematical proof #Multiplicity (mathematics) #Polynomial and algebraic computation #math.AC #msc:13A35 #msc:14B05

paper · pdf · doi:10.2140/ant.2018.12.61

published as Alg. Number Th. 12 (2018) 61-97 · Minor changes and corrections

openalex created_date 2016/09/30 · arxiv created 2016/10/28 · openalex publication_date 2018/03/13 · arxiv updated 2018/04/04 · openalex updated_date 2026/08/05

Abstract

We present a unified approach to the study of [math] -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 [math] -signature to be viewed as the infimum of relative differences in the Hilbert–Kunz multiplicities of the cofinite ideals in a local ring.

Citations

• Characterizations of Regular Local Rings of Characteristic p
• Two theorems about maximal Cohen-Macaulay modules
• Tight Closure, Invariant Theory, and the Briancon-Skoda Theorem

Cited by

• Bertini Theorems for F-signature and Hilbert-Kunz multiplicity