Betti numbers of lex ideals over some Macaulay-lex rings
Among the most important theorems on Hilbert functions of homogeneous ideals over polynomial rings are Macaulay's theorem [\textit{F. S. Macaulay}, Proceedings L. M. S. (2) 26, 531--555 (1927; JFM 53.0104.01)], and its extension by Bigatti, Hulett and Pardue [\textit{A. M. Bigatti}, Commun. Algebra 21, No. 7, 2317--2334 (1993; Zbl 0817.13007); \textit{H. A. Hulett}, Commun. Algebra 21, No. 7, 2335--2350 (1993; Zbl 0817.13006); \textit{K. Pardue}, Ill. J. Math. 40, No. 4, 564--585 (1996; Zbl 0903.13004)]. A very interesting problem is to extend these results to other rings, see for instance [\textit{V. Gasharov, N. Horwitz} and \textit{I. Peeva}, Mich. Math. J. 57, 339--357 (2008; Zbl 1183.13031)]. The present papers studies generalizations of the aforementioned results to quotient rings of the form \(A/M\) where \(A\) is a polynomial ring and \(M\) is a monomial ideal of \(A\). The main results of this paper are the following: 1. The introduction of a new class of rings, namely \textit{colored squarefree rings}, which are Macaulay-lex i.e. such that for every homogeneous ideal \(J\) of them there exists a lex ideal having the same Hilbert function as \(J\) (Definition 1.3 and Theorem 2.1). This result generalizes a theorem of \textit{P. Frankl, Z. Füredi} and \textit{G. Kalai} [Math. Scand. 63, No. 2, 169--178 (1988; Zbl 0651.05003)] which characterizes face vectors of colored simplicial complexes. 2. Most colored squarefree ideals do not admit ideals with maximal Betti numbers (Proposition 3.2 and Theorem 3.5). This disproves a conjecture of Mermin and Peeva that lex-plus-\(M\) ideals of \(A\) have greatest graded Betti numbers among all homogeneous-plus-\(M\) ideals of \(A\) having the same Hilbert function when \(M\) is a monomial ideal such that \(A/M\) is Macaulay-lex. 3. The computation of the graded Betti numbers of Borel ideals over a colored squarefree ring. This is a first step to study the behaviour of Betti numbers of lex ideals of Macaulay-lex rings. The paper finishes with a section on related open problems on Betti numbers of lex ideals, Betti numbers of Stanley-Reisner ideals and h-vectors of balanced Cohen-Macaulay complexes.
scientific article; zbMATH DE number 5703813
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