Extremal Betti Numbers and Applications to Monomial Ideals

Details Extremal Betti Numbers and Applications to Monomial Ideals Extremal Betti Numbers and Applications to Monomial Ideals

1998-04-08

arxiv.org/abs/math/9804052v2

Mathematics

Commutative Algebra

Minor revision. 15 pages, Plain TeX with epsf.tex, 8 PostScript figures, PostScript file available also at http://www.math.c

Scientific paper

In this short note we introduce a notion of extremality for Betti numbers of a minimal free resolution, which can be seen as a refinement of the notion of Mumford-Castelnuovo regularity. We show that extremal Betti numbers of an arbitrary submodule of a free S-module are preserved when taking the generic initial module. We relate extremal multigraded Betti numbers in the minimal resolution of a square free monomial ideal with those of the monomial ideal corresponding to the Alexander dual simplicial complex and generalize theorems of Eagon-Reiner and Terai. As an application we give easy (alternative) proofs of classical criteria due to Hochster, Reisner, and Stanley.

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