Green-Lazarsfeld index of square-free monomial ideals and their powers

2021/10/23 by Ghouchan, Mohammad Farrokhi Derakhshandeh, Sadegh, Yasin, Pour, Ali Akbar Yazdan
#05E40 (Primary) 13P20 (Secondary) #13D02 #Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.2110.12174

Abstract

Let \mathbbK be a field and I be a square-free monomial ideal in the polynomial ring \mathbbK[x₁, …, x_n]. 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(I^k)>1 for all k if I is any square-free monomial ideal with index(I)>1.

Related

• Candidates for non-zero Betti numbers of monomial ideals
• Free resolution of powers of monomial ideals and Golod rings
• Stanley-Reisner ideals with linear powers
• Depth and minimal number of generators of square free monomial ideals
• The normalized depth function of squarefree powers