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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
##### Keywords:
bounds for Betti numbers; Hilbert function
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##### References:
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