# zbMATH — the first resource for mathematics

Upper bounds for the Betti numbers of a given Hilbert function. (English) Zbl 0817.13007
From a paper by F. S. Macaulay [Proc. Lond. Math. Soc. 26, 531-555 (1927)] it follows that a lex segment ideal has the greatest number of generators (the 0-th Betti number $$\beta_ 0)$$ among all the homogeneous ideals with the same Hilbert function.
In this paper we prove that this fact extends to every Betti number, in the sense that all the Betti numbers of a lex segment ideal are bigger than or equal to the ones of any homogeneous ideal with the same Hilbert function.
[The same result can be found in the paper by H. A. Hulett, see the preceding review].

##### MSC:
 13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series 13D05 Homological dimension and commutative rings
##### Keywords:
bounds for Betti numbers; Hilbert function
Full Text:
##### References:
 [1] Bigatti A.M., AAECC 2 pp 21– (1991) · Zbl 0734.13016 · doi:10.1007/BF01810852 [2] Eliahou S., J. Algebra 129 pp 1– (1990) · Zbl 0701.13006 · doi:10.1016/0021-8693(90)90237-I [3] Galligo A., Lecture Notes in Mathematics 409 pp 543– (1974) [4] Green, M. 1988. Restriction of linear series to hyperplanes, and some results of Macaulay and Gotzmann. Algebraic Curves and Projective Geometry Proceedings. 1988, Trento. Vol. 1389, pp.76–86. Berlin: Springer-Verlag. Lecture Notes in Mathematiecs Heidelberg, New York [5] Macaulay F.S., Proc. London Math. Soc 26 pp 531– (1927) · JFM 53.0104.01 · doi:10.1112/plms/s2-26.1.531 [6] Möller H.M., J. Algebra 100 pp 138– (1986) · Zbl 0621.13007 · doi:10.1016/0021-8693(86)90071-2 [7] Robbiano L., Queen’s Papers in Pure and Applied Mathematics 85 pp B1– (1990)
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.