Extremal Betti Numbers and Applications to Monomial Ideals

Recall that the (Mumford-Castelnuovo) regularity of M is the least integer ρ such that for each i all free generators of Fi lie in degree ≤ i + ρ, that is βi,j = 0, for j > i + ρ. In terms of Macaulay [Mac] regularity is the number of rows in the diagram produced by the “betti” command. A Betti number βi,j 6= 0 will be called extremal if βl,r = 0 for all l ≥ i and r ≥ j + 1, that is if βi,j is the nonzero top left “corner” in a block of zeroes in the Macaulay “betti” diagram. In other words, extremal Betti numbers account for “notches” in the shape of the minimal free resolution and one of them computes the regularity. In this sense, extremal Betti numbers can be seen as a refinement of the notion of Mumford-Castelnuovo regularity. In the first part of this note we connect the extremal Betti numbers of an arbitrary submodule of a free S-module with those of its generic initial module. In the second part, which can be read independently of the first, 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. Our techniques give also a simple geometric proof of a more precise version of a recent result of Terai [Te97] (see also [FT97] for a homological reformulation and related results), generalizing Eagon and Reiner’s theorem [ER96] that a StanleyReisner ring is Cohen-Macaulay if and only if the homogeneous ideal corresponding to the Alexander dual simplicial complex has a linear resolution. We are grateful to David Eisenbud for useful discussions.