×

zbMATH — the first resource for mathematics

Hilbert functions on Cohen-Macaulay ideals with assigned generators’ degrees. (English) Zbl 1167.13305
Summary: We give information on the Hilbert function of a Cohen-Macaulay ideal \(I\) of the polynomial ring \(R=k[x_0, x_1,\dots , x_r]\) which is minimally generated by \(t\) forms of degrees \(d_1,\dots,d_t\). Mainly we deal with the codimension two case in which we show that the Dubreil bound \(t\leq d_1+1\) is a necessary and sufficient condition to have such an ideal and we give a sharp upper bound and lower bound for the Hilbert function. In codimension greater than two we give a characterization for having such an ideal and in codimension 3 we find an Hilbert function which is maximal for these ideals with \(d_1=\dots d_t=a\) and we produce a scheme which realizes such a Hilbert function.
MSC:
13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
PDF BibTeX XML Cite
Full Text: EuDML
References:
[1] D. ANICK, Thin Algebras of embedding dimension three, J. of Algebra, 100 (1986), pp. 235-259. Zbl0588.13013 MR839581 · Zbl 0588.13013
[2] M. AUBRY, Sèrie de Hilbert d’une algébre de polynomes quotient, J. of Algebra, 176 (1995), pp. 392-416. Zbl0836.13011 MR1351616 · Zbl 0836.13011
[3] A. BIGATTI - A. V. GERAMITA - J. C. MIGLIORE, Geometric consequences of extremal behaviour in a theorem of Macaulay, Trans. Amer. Math. Soc., 346 (1) (1994), pp. 203-235. Zbl0820.13019 MR1272673 · Zbl 0820.13019
[4] R. FRÖBERG, An inequality for Hilbert series of graded algebras, Math. Scand., 56 (1985), pp. 117-144. Zbl0582.13007 MR813632 · Zbl 0582.13007
[5] A. FRÖBERG - J. HOLLMAN, Hilbert series for ideals generated by generic forms, J. Symbolic Comput., 17 (1994), pp. 149-157. Zbl0811.13006 MR1283740 · Zbl 0811.13006
[6] R. HARTSHORNE, Some examples of Gorenstein liaison in codimension three, Collect. Math., 53 (1) (2002), pp. 21-48. Zbl1076.14065 MR1893306 · Zbl 1076.14065
[7] M. HOCHSTER - D. LAKSOV, The linear syzygies of generic forms, Comm. Algebra, 15 (1987), pp. 227-239. Zbl0619.13007 MR876979 · Zbl 0619.13007
[8] C. HUNEKE - B. ULRICH, The structure of linkage, Ann. of Math., 126 (2) (1987), pp. 277-334. Zbl0638.13003 MR908149 · Zbl 0638.13003
[9] O. KLEPPE - J. MIGLIORE - R. MIRÒ ROIG - U. NAGEL - C. PETERSON, Gorenstein liaison, complete intersection liaison invariants and unobstructedness, Mem. Amer. Math. Soc., 154 (2001). Zbl1006.14018 MR1848976 · Zbl 1006.14018
[10] J. MIGLIORE - R. MIRÒ ROIG, On the minimal free resolution of n1 11 generic forms, Trans. Amer. Math. Soc., 355, no. 1 (2003), pp. 1-36. Zbl1053.13005 MR1928075 · Zbl 1053.13005
[11] J. MIGLIORE - U. NAGEL, Lifting monomial ideals, Comm. in Alg., 28 (12) (2000), pp. 5679-5701. Zbl1003.13005 MR1808596 · Zbl 1003.13005
[12] A. RAGUSA - G. ZAPPALÀ, Partial intersection and graded Betti numbers, Beiträge zur Algebra und Geometrie, 44, no. 1 (2003), pp. 285-302. Zbl1033.13004 MR1991000 · Zbl 1033.13004
[13] R. STANLEY, Weyl groups, the hard Lefschetz theorem, and the Sperner property, SIAM J. Algebraic Discrete Methods, 1 (1980), pp. 168-184. Zbl0502.05004 MR578321 · Zbl 0502.05004
[14] R. STANLEY, Hilbert functions of graded algebras, Adv. in Math., 28 (1978), pp. 57-83. Zbl0384.13012 MR485835 · Zbl 0384.13012
[15] J. WATANABE, the Dilworth number of Artinian rings and finite posets with rank function, Commutative Algebra and Combinatorics, Advanced Studies in Pure Math. Vol. 11, Kinokuniya Co. North Holland, Amsterdam (1987), pp. 303-312. Zbl0648.13010 MR951211 · Zbl 0648.13010
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.