×

On the depth of the symmetric algebra. (English) Zbl 0604.13008

Let (A,\({\mathfrak n})\) be a regular local ring with an infinite residue class field and let (R,\({\mathfrak m})\) be the local ring A/I, I an ideal of A such that \(I\subset {\mathfrak n}^ 2\). Let S(\({\mathfrak m})\) be the symmetric algebra of \({\mathfrak m})\). In the main theorem of this paper, which is given in section \(2,\) depth gr\({}_ n(I)\) and depth S(m) are compared. Let \(d=depth R;\)
(1) If depth gr\({}_ n(I)\leq d\), then depth S(m)\(=depth gr_ n(I).\)
(2) If depth gr\({}_ n(I)=d+1\), then depth S(m)\(\geq d.\)
Another part of the theorem deals with the question, when is depth S(m)\(\geq d+1 ?\) For this the authors introduce the notion of a strong socle. The depth of \(gr_ n(I)\) is studied in section \(3\) and the strong socle condition is discussed in section \(4.\) In section \(5\) depth S(m) is computed for a class of local rings.
Reviewer: M.-A.Knus

MSC:

13C15 Dimension theory, depth, related commutative rings (catenary, etc.)
13H05 Regular local rings
13D25 Complexes (MSC2000)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. · Zbl 0175.03601
[2] Michela Brundu, Normal flatness and isomultiplicity, Rend. Sem. Mat. Univ. Politec. Torino 40 (1982), no. 1, 163 – 172 (Italian, with English summary). · Zbl 0535.14008
[3] David Eisenbud and Shiro Goto, Linear free resolutions and minimal multiplicity, J. Algebra 88 (1984), no. 1, 89 – 133. · Zbl 0531.13015 · doi:10.1016/0021-8693(84)90092-9
[4] R. Gebauer and H. Kredel, Buchberger’s algorithm for constructing canonical bases (Gröbner bases) for polynomial ideals, Program Documentation, Univ. Heidelberg, 1983.
[5] J. Herzog and M. Kühl, On the Betti numbers of finite pure and linear resolutions, Comm. Algebra 12 (1984), no. 13-14, 1627 – 1646. · Zbl 0543.13008 · doi:10.1080/00927878408823070
[6] J. Herzog, A. Simis, and W. V. Vasconcelos, Koszul homology and blowing-up rings, Commutative algebra (Trento, 1981) Lecture Notes in Pure and Appl. Math., vol. 84, Dekker, New York, 1983, pp. 79 – 169. · Zbl 0499.13002
[7] C. Huneke and M. Rossi, The dimension and components of symmetric algebras, J. Algebra 98 (1986), no. 1, 200 – 210. · Zbl 0584.13010 · doi:10.1016/0021-8693(86)90023-2
[8] L. Robbiano, Coni tangenti a singolarita razionali, Curve Algebriche, Istituto di Analisi Globale, Firenze, 1981.
[9] Lorenzo Robbiano and Giuseppe Valla, Free resolutions for special tangent cones, Commutative algebra (Trento, 1981) Lecture Notes in Pure and Appl. Math., vol. 84, Dekker, New York, 1983, pp. 253 – 274. · Zbl 0558.14008
[10] Lorenzo Robbiano and Giuseppe Valla, On the equations defining tangent cones, Math. Proc. Cambridge Philos. Soc. 88 (1980), no. 2, 281 – 297. · Zbl 0469.14001 · doi:10.1017/S0305004100057583
[11] M. E. Rossi, Sulle algebre di Rees e simmetrica di un ideale, Le Matematiche 34 (1979), 1-2. · Zbl 0513.13001
[12] Judith D. Sally, On the associated graded ring of a local Cohen-Macaulay ring, J. Math. Kyoto Univ. 17 (1977), no. 1, 19 – 21. · Zbl 0353.13017
[13] Peter Schenzel, Über die freien Auflösungen extremaler Cohen-Macaulay-Ringe, J. Algebra 64 (1980), no. 1, 93 – 101 (German). · Zbl 0449.13008 · doi:10.1016/0021-8693(80)90136-2
[14] Paolo Valabrega and Giuseppe Valla, Form rings and regular sequences, Nagoya Math. J. 72 (1978), 93 – 101. · Zbl 0362.13007
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.