Abstract
Let K be a field and I be a square-free monomial ideal in the polynomial ring K[x1,…,xn]. The Green-Lazarsfeld index, index(I), counts the number of steps to reach to a syzygy minimally generated by a nonlinear form in a graded minimal free resolution of I. In this paper, we study this invariant for I and its powers from a combinatorial point of view. We characterize all square-free monomial ideals I generated in degree 3 such that index(I)>1. Utilizing this result, we also characterize all square-free monomial ideals generated in degree 3 such that index(I)>1 and index(I2)=1. In case n≤5, it is shown that index(Ik)>1 for all k if I is any square-free monomial ideal with index(I)>1.
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