Irreducible affine space curves and the uniform Artin-Rees property on the prime spectrum
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.
scientific article; zbMATH DE number 5378370
scientific article