arXiv:2110.12174v1 [math.AC] 23 Oct 2021 GREEN-LAZARSFELD INDEX OF SQUARE-FREE MONOMIAL IDEALS AND THEIR POWERS MOHAMMAD FARROKHI D. G., YASIN SADEGH, AND ALI AKBAR YAZDAN POUR 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(I 2 ) = 1. In case n ≤ 5, it is shown that index(I k ) > 1 for all k if I is any square-free monomial ideal with index(I ) > 1. Introduction Let K be a field and S = K[x 1 ,...,x n ] be the polynomial ring in n variables endowed with the standard grading (i.e. deg(x i ) = 1 for all 1 ≤ i ≤ n). Let I = 0 be a homogeneous ideal in S generated by elements of degree d. A classical problem in algebraic geometry and commutative algebra is to study equations of syzygies of I and determine when these equations are linear forms. The ideal I is said to satisfy the N d,p -property (p> 0), if β i,i+j (I )=0, for all i < p, and j > d. The quantity index(I ) = sup {p : I satisfies N d,p -property} is called the Green-Lazarsfeld index, or simply index of I . The N d,p notation defined in [10] comes essentially from the notation N p of Green and Lazarsfeld in [15, 14] (see also [8]). This notation indicates when the minimal equations defining the syzygies are in the simplest form (that is of the linear form). The Green-Lazarsfeld index of ideals is very difficult to compute in general. Important conjectures, such as Green’s conjecture [9, Chapter 9], predicts the value of this invariant for certain families of varieties. In [4] the authors study the Green-Lazarsfeld index of the Veronese embeddings ν c : P n−1 → P N of degree c of projective spaces and, more generally, of the Veronese embeddings of arbitrary varieties (see also [13, 17, 20, 21, 23] in this matter). In their interesting paper [10], Eisenbud, Green, Hulek, and Popescu provide some nice examples and conjectures about N 2,p -property. As a remarkable result, they characterize the property N 2,p for monomial ideals in degree 2 [10, Theorem 2.1, Proposition 2.3]. It turns out that the edge ideal I (G) of a graph G satisfies N 2,p -property if and only if every induced cycle in ¯ G has length ≥ p + 3. In particular, index(I (G)) = inf  |C |− 3: C is an induced cycle in ¯ G of length > 3  . 2010 Mathematics Subject Classification. Primary: 13D02, 05E40; Secondary: 13P20. Key words and phrases. Green-Lazarsfeld index, linear syzygy, powers of ideal, clutter. 1