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Maximum Betti numbers of homogeneous ideals with a given Hilbert function. (English) Zbl 0817.13006
Definition. Monomials \(m_{i_ 1}, m_{i_ 2}, \dots, m_{i_ r}\) of degree \(d\) will be called a lexicographic (lex) segment if \(m_{i_ 1} > m_{i_ 2} > \cdots > m_{i_ r}\) in the lexicographic order and this list is saturated. Definition. \(I\) is called a lexicographic segment ideal if one can pick a set of generators such that the generators of \(I\) of minimum degree \(d_ 0\) form a lexicographic segment beginning with \(x_ 1^{d_ 0}\), \(x_ 1^{d_ 0-1} x_ 2, \dots\) and the generators of \(I\) in degree \(d \geq d_ 0\) form a lexicographic segment starting with the largest monomial of degree \(d\) not divisible by a generator of lower degree.
Theorem (Macaulay). Among all homogeneous ideals having a given Hilbert function, the lex segment ideal will always have the largest number of generators.
In this paper, we generalize this result. Main theorem:
Let \(I\) be the lex segment ideal with a given Hilbert function \(H\) and let \(J\) be any other homogeneous ideal with \(H_{(R/J)} = H_{(R/I)} = H\). Then for each \(i\), \(\beta_ i (R/I) \geq \beta_ i (R/J)\) for every \(i\) \((\beta_ i = i\)-th Betti number).
[The same result can be found in the paper by A. M. Bigatti, see the following review)].

MSC:
13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
13D05 Homological dimension and commutative rings
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[1] Avramov L., American Journal of Mathematics 103 pp 1– (1981) · Zbl 0447.13006 · doi:10.2307/2374187
[2] Bayer D., Ph D. Thesis (1982)
[3] Bayer D., Duke Mathematical Journal 55 pp 321– (1987) · Zbl 0638.06003 · doi:10.1215/S0012-7094-87-05517-7
[4] Glements G., Journal of Comhinatorial Theory 7 pp 230– (1969) · Zbl 0186.01704 · doi:10.1016/S0021-9800(69)80016-5
[5] Eisenbud D., Commuatative Algebra with a view toward Algebraic Geometry (1992)
[6] Gallîigo A., Fonctions de Plusieurs Variables Complexes. Lect. Notes Math 409 pp 543– (1974) · doi:10.1007/BFb0068121
[7] Eliahou S., Journal of Algebra 129 pp 1– (1990) · Zbl 0701.13006 · doi:10.1016/0021-8693(90)90237-I
[8] Macaulay F.S., Proc. London Math. Soc 26 pp 531– (1927) · JFM 53.0104.01 · doi:10.1112/plms/s2-26.1.531
[9] Taylor, D. 1966. ”PhD. Thesis”. University of Chicago.
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