×

zbMATH — the first resource for mathematics

Betti numbers, Castelnuovo Mumford regularity, and generalisations of Macaulay’s theorem. (English) Zbl 0908.13007
Let \(I(H)\) be the set of homogeneous ideals in \(k[x_0, \dots, x_n]\) with Hilbert function \(H\). Then the set of sequences of Betti numbers, \(\beta (I(H))= \{(\beta_0 (I), \dots, \beta_l(I))\); \(I\in I(H)\}\), is finite. The set \(\beta (I(H))\) is partially ordered by \((a_0, \dots,a_l) \leq(b_0, \dots, b_l)\) if \(a_i\leq b_i\) for all \(i\). Bigatti and Hulett have independently proved that \(\beta (I(H))\) has a unique maximal element which corresponds to the lexicographical monomial ideal in \(I(H)\).
In this paper the subsets \(I(H,m)\) and \(IS(H,m)\) of \(I(H)\), consisting of homogeneous (saturated homogeneous, resp.) ideals with Castelnuovo-Mumford regularity at most \(m\) are considered. It is shown that, in characteristic 0, \(\beta (I(H,m))\) and \(\beta (IS(H,m))\) contain maximal elements and these are calculated.

MSC:
13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
13D02 Syzygies, resolutions, complexes and commutative rings
13D03 (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Atiah M.F., Introduciton to Commutative Algebra. (1969)
[2] Bayer D.A., The division algorithm and the Hilbert scheme (1982)
[3] Boratynski M.S., The Curves Seminar at Queen’s 3 (1984)
[4] Bruns W.J., Cohen-Macaulay rings,39 of Combridge studies in advanced mathematics (1993)
[5] Bigatti A.M., Communications in Albegra 21 pp 2317– (1993) · Zbl 0817.13007 · doi:10.1080/00927879308824679
[6] Bayer D.A., what can be computed in algebraic geometry (1993) · Zbl 0846.13017
[7] Bayer D.A., Invent, Math 87 pp 1– (1987) · Zbl 0625.13003 · doi:10.1007/BF01389151
[8] Eisenbud D., Commutative Algebra with a View Toward Algebraic Geometry (1995) · Zbl 0819.13001
[9] Eliahou S., Journal of Algebra 129 pp 1– (1990) · Zbl 0701.13006 · doi:10.1016/0021-8693(90)90237-I
[10] Galligo A., Ann.Inst. Fourier 29 pp 107– (1979) · Zbl 0412.32011 · doi:10.5802/aif.745
[11] Gotzmann G., Math.z 158 pp 61– (1978) · Zbl 0352.13009 · doi:10.1007/BF01214566
[12] Hartshorne R., Publ. Math. de I.H.E.S 29 pp 261– (1966)
[13] Hulett H.A., Communications in Algebra 21 pp 2335– (1993) · Zbl 0817.13006 · doi:10.1080/00927879308824680
[14] Hulett H.A., Communications in Algebra 23 pp 1249– (1995) · Zbl 0816.13012 · doi:10.1080/00927879508825278
[15] Mall D., submitted 23 (1996)
[16] Moller M., J. Alg., 100 pp 138– (1986) · Zbl 0621.13007 · doi:10.1016/0021-8693(86)90071-2
[17] Pardue K., Nonstandard Borel-Fixed Ideals (1994)
[18] Renschuch B., Elementare and praktische Idealtheorie (1976)
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.