2025 Colength, Multiplicity, and Ideal Closure Operations II
Linquan Ma, Pham Hung Quy, Ilya Smirnov
Michigan Math. J. Advance Publication 1-30 (2025). DOI: 10.1307/mmj/20236391

Abstract

Let (R,m) be a Noetherian local ring. This paper concerns several extremal invariants arising from the study of the relation between colength and (Hilbert–Samuel or Hilbert–Kunz) multiplicity of an m-primary ideal. We introduce versions of these invariants by restricting to various closures and “cross-pollinate” the two multiplicity theories by asking for analog invariants already established in one of the theories.

On the Hilbert–Samuel side, we prove that the analog of the Stückrad–Vogel invariant (the infimum of the ratio between the multiplicity and colength) for integrally closed m-primary ideals is often 1 under mild assumptions. We also compute the supremum and infimum of the relative drops of multiplicity for (integrally closed) m-primary ideals. On the Hilbert–Kunz side, we study several analogs of the Lech–Mumford and Stückrad–Vogel invariants.

Dedication

Dedicated to Professor Watanabe on the occasion of his 80th birthday

Citation

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Linquan Ma. Pham Hung Quy. Ilya Smirnov. "Colength, Multiplicity, and Ideal Closure Operations II." Michigan Math. J. Advance Publication 1 - 30, 2025. https://doi.org/10.1307/mmj/20236391

Information

Received: 30 May 2023; Revised: 11 August 2024; Published: 2025
First available in Project Euclid: 23 January 2025

Digital Object Identifier: 10.1307/mmj/20236391

Keywords: 13A35 , 13B22 , 13D40 , 13H15

Rights: Copyright © 2024 The University of Michigan

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