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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)].

13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
13D05 Homological dimension and commutative rings
Full Text: DOI
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