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Analytic spread and non-vanishing of asymptotic depth

Published online by Cambridge University Press:  08 March 2017

CLETO B. MIRANDA–NETO
CLETO B. MIRANDA–NETO*
Affiliation:
Departamento de Matemática, Universidade Federal da Paraíba, 58051-900 João Pessoa, PB, Brazil. e-mail: cleto@mat.ufpb.br
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Abstract

Let S be a polynomial ring over a field K of characteristic zero and let M ⊂ S be an ideal given as an intersection of powers of incomparable monomial prime ideals (e.g., the case where M is a squarefree monomial ideal). In this paper we provide a very effective, sufficient condition for a monomial prime ideal P ⊂ S containing M be such that the localisation MP has non-maximal analytic spread. Our technique describes, in fact, a concrete obstruction for P to be an asymptotic prime divisor of M with respect to the integral closure filtration, allowing us to employ a theorem of McAdam as a bridge to analytic spread. As an application, we derive – with the aid of results of Brodmann and Eisenbud-Huneke – a situation where the asymptotic and conormal asymptotic depths cannot vanish locally at such primes.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 163 , Issue 2 , September 2017 , pp. 289 - 299
Copyright
Copyright © Cambridge Philosophical Society 2017 

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