Candidates for nonzero Betti numbers of monomial ideals

摘要

Let I be a monomial Ideal in the polynomial ring S generated by elements of degree at most d. In this paper, it is shown that, if the i-th syzygy of l has no elements of degrees j, ...,j + (d - 1) (where j >= i + d), then (i + 1)-th syzygy of i does not have any element of degree j + d. Then we give several applications this result, including an alternative proof for Green-Lazarsfeld index of the edge ideals of graphs as well as an alternative proof for Froberg's theorem on classification of square-free monomial ideals generated in degree 2 with linear resolution. Among all, we deduce a partial result on subadditivity of the syzygies for monomial ideals