×

The strong Rees property of powers of the maximal ideal and Takahashi-Dao’s question. (English) Zbl 1471.13012

For a local noetherian ring \((A,\mathfrak{m})\), the concept of \(\mathfrak{m}\)-full ideal was introduced by D. Rees and J. Watanabe [Nagoya Math. J. 106, 101–111 (1987; Zbl 0623.13012)]. An \(\mathfrak{m}\)-primary ideal \(I\) is said to be \(\mathfrak{m}\)-full if there exists \(x \in \mathfrak{m}\) such that \((\mathfrak{m} I : x)=I\). They also proved that these ideals satisfy the “Rees property”; that is, for every ideal \(J\) containing \(I\) we have \(\mu(J) \leq \mu(I)\), where \(\mu(I)\) denotes the minimal number of generators of \(I\).
Motivated by an inequality of H. Dao relating the minimal number of generators \(\mu(I)\), the multiplicity \(e(I)\), and the Loewy length \(\ell\ell(I)=\min\{ n \mid \mathfrak{m}^n \subseteq I \}\), in this paper the authors introduce the notion of “strong Rees property” for an \(\mathfrak{m}\)-primary ideal \(I\) by requiring \(\mu(J) < \mu(I)\) for every ideal \(J\) strictly containing \(I\). If the depth of the associated graded ring \(G_\mathfrak{m}(A)\) is at least two, the authors prove that all the powers of the maximal ideal satisfy the strong Rees property, and, as an application, are able to answer a question of Takahashi and Dao. If \((A,\mathfrak{m})\) is a two-dimensional excellent normal local domain containing an algebraically closed field, then the following are equivalent:
(1)
the Rees algebra of the ideal \(\mathfrak{m}\) is a Cohen-Macaulay normal domain;
(2)
the inequality \((\mu(I)-1)\ell\ell(I) \geq e(I)\) holds for every \(\mathfrak{m}\)-primary integrally closed ideal \(I\);
(3)
the inequality \((\mu(I)-1)\ell\ell(I) \geq e(I)\) holds for every power \(I=\mathfrak{m}^r\) of the maximal ideal.

MSC:

13A30 Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics
13H15 Multiplicity theory and related topics
13B22 Integral closure of commutative rings and ideals
14B05 Singularities in algebraic geometry

Citations:

Zbl 0623.13012
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Asadollahi, J.; Puthenpurakal, T. J., An analogue of a theorem due to Levin and Vasconcelos, (Commutative Algebra and Algebraic Geometry. Commutative Algebra and Algebraic Geometry, Contemp. Math., vol. 390 (2005), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 9-15 · Zbl 1191.13016
[2] Dao, H.; Smirnov, I., The multiplicity and the number of generators of an integrally closed ideal
[3] Goto, S., Integral closedness of complete intersection ideals, J. Algebra, 108, 151-160 (1987) · Zbl 0629.13004
[4] Goto, S.; Iai, S.; Watanabe, K.-i., Good ideals in Gorenstein local rings, Trans. Amer. Math. Soc., 353, 6, 2309-2346 (2001), (electronic) · Zbl 0966.13002
[5] Goto, S.; Watanabe, K.-i., On graded rings, I, J. Math. Soc. Japan, 30, 179-213 (1978) · Zbl 0371.13017
[6] Goto, S.; Takahashi, R.; Taniguchi, N., Almost Gorenstein rings – towards a theory of higher dimension, J. Pure Appl. Algebra, 219, 2666-2712 (2015) · Zbl 1319.13017
[7] Herzog, J., When is a regular sequence super regular?, Nagoya Math. J., 83, 183-195 (1981) · Zbl 0431.13015
[8] Huckaba, S.; Huneke, C., Normal ideals in regular rings, J. Reine Angew. Math., 510, 63-82 (1999) · Zbl 0923.13005
[9] Komeda, J., On the existence of Weierstrass points with a certain semigroup generated by 4 elements, Tsukuba J. Math., 6, 2, 237-270 (1982) · Zbl 0546.14011
[10] Komeda, J.; Ohbuchi, A., Existence of the non-primitive Weierstrass gap sequences on curves of genus 8, Bull. Braz. Math. Soc. (N.S.), 39, 109-121 (2008) · Zbl 1133.14307
[11] Robbiano, L.; Valla, G., On the equations defining tangent cones, Math. Proc. Cambridge Philos. Soc., 88, 281-297 (1980) · Zbl 0469.14001
[12] Okuma, T.; Watanabe, K.-i.; Yoshida, K., Good ideals and \(p_g\)-ideals in two-dimensional normal singularities, Manuscripta Math., 150, 499-520 (2016) · Zbl 1354.13011
[13] Okuma, T.; Watanabe, K.-i.; Yoshida, K., Rees algebras and \(p_g\)-ideals in a two-dimensional normal local domain, Proc. Amer. Math. Soc., 145, 1, 39-47 (2017) · Zbl 1357.13011
[14] Puthenpurakal, T. J., Ratliff-Rush filtration, regularity and depth of higher associated graded modules. I, J. Pure Appl. Algebra, 208, 1, 159-176 (2007) · Zbl 1106.13003
[15] Puthenpurakal, T. J., Ratliff-Rush filtration, regularity and depth of higher associated graded modules. Part II, J. Pure Appl. Algebra, 221, 3, 611-631 (2017) · Zbl 1349.13010
[16] Ratliff, L. J.; Rush, D. E., Two notes on reduction of ideals, Indiana Univ. Math. J., 27, 6, 929-934 (1978) · Zbl 0368.13003
[17] Sally, J. D., On the associated graded ring of a local Cohen-Macaulay ring, J. Math. Kyoto Univ., 17, 19-21 (1977) · Zbl 0353.13017
[18] Sally, J. D., Cohen-Macaulay local rings of maximal embedding dimension, J. Algebra, 56, 168-183 (1979) · Zbl 0401.13016
[19] Sally, J. D., Cohen-Macaulay local rings of embedding dimension \(e + d - 2\), J. Algebra, 83, 393-408 (1983) · Zbl 0517.13013
[20] Dao, H.; Takahashi, R., Upper bounds for dimensions of singularity categories, C. R. Acad. Sci. Paris, Ser. I, 353, 297-301 (2015) · Zbl 1312.13021
[21] Watanabe, J., \( \mathfrak{m} \)-Full ideals, Nagoya Math. J., 106, 101-111 (1987) · Zbl 0623.13012
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.