Báo cáo toán học: "Optimal Betti numbers of forest ideals"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Optimal Betti numbers of forest ideals. | Optimal Betti numbers of forest ideals Michael Goff Department of Mathematics University of Washington Seattle WA 98195-4350 USA mgoff@ Submitted Dec 19 2008 Accepted Mar 3 2009 Published Mar 13 2009 Mathematics Subject Classification 05E99 13D02 Abstract We prove a tight lower bound on the algebraic Betti numbers of tree and forest ideals and an upper bound on certain graded Betti numbers of squarefree monomial ideals. 1 Introduction In this paper we consider bounds on the algebraic Betti numbers of squarefree monomial ideals. These ideals are naturally related both to hypergraphs and to simplicial complexes and understanding the structure of their minimal free resolutions leads to insights into the combinatorics of hypergraphs and simplicial complexes. For example the f-vector of a simplicial complex can be expressed as alternating sums of certain algebraic Betti numbers. Several other papers including 7 10 and the survey paper 12 use combinatorial methods to describe the minimal free resolutions of edge ideals and to bound their Betti numbers. For example Ferrers ideals as described in 4 and 5 are conjectured in 17 and shown in 9 to minimize Betti numbers among edge ideals of bipartite graphs. Earlier papers construct bounds on Betti numbers in terms of the projective dimension 2 or the Hilbert function 1 . In general while constructing explicit generally nonminimal resolutions such as the Taylor resolution is effective in finding upper bounds on Betti numbers there are no standard techniques for finding lower bounds. One important lower bound on the Betti numbers is the Buchsbaum-Eisenbud-Horrocks conjecture which states that for a graded module with projective dimension l and Krull dimension 0 the i-th Betti number is at least hi M i . This was proven in 3 for a class of modules that includes all finite length quotients of S by monomial ideals and 8 proves a version of this conjecture for modules of monomial type over local rings. In this .

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