Irreducible affine space curves and the uniform Artin-Rees property on the prime spectrum
type: Item
instance of: scholarly article
title: Irreducible affine space curves and the uniform Artin-Rees property on the prime spectrum
zbMATH Open document ID: 1152.13013
author: Liam O'Carroll
author: Francesc Planas-Vilanova
published in: Journal of Algebra
publication date: 2008-12-05T00:00:00Z
review text: Let \(A\) be a noetherian ring, \(I \subset A\) an ideal, and let \(N \subset M\) be finitely generated \(A\)-modules. By the Artin-Rees Lemma there exists an integer \(s\) such that \(I^nM \cap N = I^{n-s}(I^sM \cap N)\) for all \(n \geq s.\) One says that the pair \((N,M)\) has the Artin-Rees property with respect to the class \(\mathcal C\) if this equality holds for all \(I \in {\mathcal C}\) with a fixed number \(s\). \textit{A. J. Duncan} and the first author [see J. Reine Angew. Math. 394, 203--207 (1989; Zbl 0659.13002)] have shown that the uniform Artin-Rees property holds for the class \(\mathcal C\) of maximal ideals of an excellent ring \(A.\) In the frame of their consideration they posed the question whether the uniform Artin-Rees property holds for the class \(\mathcal C\) of prime ideals in a noetherian ring \(A\). Note that this holds locally by the previous results. The main result of the present paper is a negative answer to the uniform Artin-Rees property on the prime spectrum. To this end the author study irreducible monomial affine space curves in \(\mathbb A_k^3\), \(k\) a field, parametrized by \(X_1 = t^{n_1}, X_2 = t^{n_2}, X_3 = t^{n_3},\) for certain integers \(n_1, n_2, n_3\) forming a one parameter family.
reviewed by: Peter Schenzel
Mathematics Subject Classification ID: 13E05
Mathematics Subject Classification ID: 14H20
zbMATH DE Number: 5378370
zbMATH Keywords: uniform Artin-Rees property
zbMATH Keywords: relation type
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2008.06.029
OpenAlex ID: W2089387686
cites work: A full uniform Artin-Rees theorem.
cites work: Q3972815
cites work: A Nullstellensatz with nilpotents and Zariski's Main Lemma on holomorphic functions
cites work: Q5670698
cites work: Integral degree of a ring, reduction numbers and uniform Artin-Rees numbers
cites work: Q3203119
cites work: Generators and relations of abelian semigroups and semigroup rings
cites work: Uniform bounds in noetherian rings
cites work: Desingularizations and the Uniform Artin-Rees Theorem
cites work: Q3676243
cites work: Sur les algèbres universelles
cites work: Q5824413
cites work: A homological investigation of a certain residual ideal
cites work: A note on Artin-Rees Numbers
cites work: The strong uniform Artin-Rees property in codimension one
cites work: Normal Transversality and Uniform Bounds
cites work: Hilbert functions, Castelnuovo-Mumford regularity and uniform Artin-Rees numbers
cites work: Q3929880
cites work: Computational methods of commutative algebra and algebraic geometry. With chapters by David Eisenbud, Daniel R. Grayson, Jürgen Herzog and Michael Stillman
cites work: Some uniform properties of 2-dimensional local rings
cites work: Q5732722
DOI: 10.1016/J.JALGEBRA.2008.06.029
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description: scientific article; zbMATH DE number 5378370