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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].

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