On Fiber Cones of $\text{m}$ -Primary Ideals

Jayanthan, A. V. ; Puthenpurakal, Tony J. ; Verma, J. K.

Canadian journal of mathematics, Tome 59 (2007) no. 1, pp. 109-126

Résumé

Two formulas for the multiplicity of the fiber cone $F\left( I \right)\,=\,\oplus _{n=0}^{\infty }{{I}^{n}}/\text{m}{{I}^{n}}$ of an $\text{m}$ -primary ideal of a $d$ -dimensional Cohen–Macaulay local ring $\left( R,m \right)$ are derived in terms of the mixed multiplicity ${{e}_{d-1}}\left( \text{m }|I \right)$ , the multiplicity $e\left( I \right)$ , and superficial elements. As a consequence, the Cohen–Macaulay property of $F\left( I \right)$ when $I$ has minimal mixed multiplicity or almost minimal mixed multiplicity is characterized in terms of the reduction number of $I$ and lengths of certain ideals. We also characterize the Cohen–Macaulay and Gorenstein properties of fiber cones of $\text{m}$ -primary ideals with a $d$ -generated minimal reduction $J$ satisfying $\ell \left( {{I}^{2}}/JI \right)\,=\,1$ or $l\left( Im \right)Jm) = 1$ .

Détail

DOI : 10.4153/CJM-2007-005-8

Mots-clés : 13H10, 13H15, 13A30, 13C15, 13A02, fiber cones, mixed multiplicities, joint reductions, Cohen–Macaulay fiber cones, Gorenstein fiber cones, ideals having minimal and almost minimal mixed multiplicities

Bibliographie

[1] [1] Bhattacharya, P. B., The Hilbert function of two ideals. Proc. Cambridge Philos. Soc. 53(1957), 568–575. Google Scholar

[2] [2] Bruns, W. and Herzog, J., Cohen–Macaulay Rings. Cambridge Studies in Advanced Mathematics 39. Cambridge University Press, Cambridge, 1993. Google Scholar

[3] [3] Chuai, J., Generalized parameter ideals in local C-M rings. Algebra Colloq. 3(1996), no. 3, 213–216. Google Scholar

[4] [4]CoCoATeam, CoCoA: A System for Doing Computations in Commutative Algebra. Available at http://cocoa.dima.unige.it. Google Scholar

[5] [5] Cortadellas, T. and Zarzuela, S., On the depth of the fiber cone of filtrations. J. Algebra 198(1997), no. 2, 428–445. Google Scholar

[6] [6] D’Cruz, C., Raghavan, K. N. and Verma, J. K., Cohen–Macaulay Fiber Cones. In: Commutative Algebra, Algebraic Geometry and Computational Methods. Springer-Verlag, Singapore, 1999, pp. 233–246. Google Scholar

[7] [7] D’Cruz, C. and Verma, J. K., Hilbert series of fiber cones of ideals with almost minimal mixed multiplicity. J. Algebra 251(2002), no. 1, 98–109. Google Scholar

[8] [8] Goto, S., Buchsbaumness in Rees algebras associated to ideals of minimal multiplicity. J. Algebra 213(1999), no. 2, 604–661. Google Scholar

[9] [9] Goto, S., Cohen–Macaulayness and negativity of a-invariants in Rees algebras associated to m-primary ideals of minimal multiplicity. J. Pure Appl. Algebra 152(2000), no. 1-3, 93–107. Google Scholar

[10] [10] Goto, S. and Shimoda, Y., On the Rees algebras of Cohen–Macaulay local rings. In: Commutative Algebra, Lecture Notes in Pure and Applied Mathematics, 68, Marcel Dekker, New York 1982, pp. 201–231. Google Scholar

[11] [11] Hyry, E., The diagonal subring and the Cohen–Macaulay property of a multigraded ring. Trans. Amer. Math. Soc. 351(1999), no. 6, 2213–2232. Google Scholar

[12] [12] Hyry, E., Cohen–Macaulay multi-Rees algebras, Compositio Math. 130(2002), no. 3, 319–343. Google Scholar

[13] [13] Jayanthan, A. V. and Verma, J. K., Hilbert coefficients and depth of fiber cones. J. Pure Appl. Algebra 201(2005), no. 1-3, 97–115. Google Scholar

[14] [14] Jayanthan, A. V. and Verma, J. K., Fiber cones of ideals of almost minimal multiplicity. Nagoya Math. J. 177(2005), 155–179. Google Scholar

[15] [15] Katz, D. and Verma, J. K., Extended Rees algebras and mixed multiplicities. Math. Z. 202(1989), no. 1, 111–128. Google Scholar

[16] [16] Northcott, D. G. and Rees, D., Reductions of ideals in local rings. Proc. Cambridge Philos. Soc. 50(1954), 145–158. Google Scholar

[17] [17] Ooishi, A., On the Gorenstein property of the associated graded ring and the Rees algebra of an ideal. J. Algebra 155(1993), no. 2, 397–414. Google Scholar

[18] [18] Rees, D., a-transforms of local rings and a theorem on multiplicities of ideals. Proc. Cambridge Philos. Soc. 57(1961), 8–17. Google Scholar

[19] [19] Rees, D., Generalizations of reductions and mixed multiplicities. J. London Math. Soc. (2) 29(1984), no. 3, 397–414. Google Scholar

[20] [20] Rossi, M. E., A bound on the reduction number of a primary ideal. Proc. Amer. Math. Soc. 128(2000), no. 5, 1325–1332. Google Scholar

[21] [21] Rossi, M. E. and Valla, G., A conjecture of J. Sally. Comm. Algebra 24(1996), no. 13, 4249–4261. Google Scholar

[22] [22] Sally, J. D., On the associated graded ring of a local Cohen–Macaulay ring. J. Math. Kyoto Univ. 17(1977), no. 1, 19–21. Google Scholar

[23] [23] Sally, J. D., Tangent cones at Gorenstein singularities. Compositio Math. 40(1980), no. 2, 167–175. Google Scholar

[24] [24] Sally, J. D., Cohen–Macaulay local rings of embedding dimension e + d –2. J. Algebra 83(1983), no. 2, 393–408. Google Scholar

[25] [25] Shah, K., On the Cohen–Macaulayness of the fiber cone of an ideal. J. Algebra 143(1991), no. 1, 156–172. Google Scholar

[26] [26] Stanley, R. P. Hilbert functions of graded algebras. Advances in Math. 28(1978), no. 1, 57–83. Google Scholar

[27] [27] Swanson, I., Tight Closure, Joint Reductions and Mixed Multiplicities. Ph. D. Thesis, Purdue University, 1992. Google Scholar

[28] [28] Teissier, B., Cycles évanescents, sections planes, et conditions de Whitney. In: Singularitiés á Cargése, Astérisque 7-8, Soc. Math. France, Paris, 1973, pp. 285–362. Google Scholar

[29] [29] Wang, Hsin-Ju, On Cohen–Macaulay local rings of embedding dimension e+d-2. J. Algebra 190(1997), no. 1, 226–240. Google Scholar