Green-Lazarsfeld index of square-free monomial ideals and their powers
Mohammad Farrokhi Derakhshandeh Ghouchan, Yasin Sadegh, and Ali Akbar, Yazdan Pour

TL;DR
This paper investigates the Green-Lazarsfeld index of square-free monomial ideals and their powers, providing combinatorial characterizations for ideals generated in degree 3 and analyzing the index behavior for small numbers of variables.
Contribution
It offers new combinatorial characterizations of degree 3 square-free monomial ideals with specific Green-Lazarsfeld index properties and examines the index behavior of their powers.
Findings
Characterization of degree 3 square-free monomial ideals with index > 1
Conditions for ideals with index > 1 and their squares having index 1
Index > 1 for all powers when n ≤ 5 and initial index > 1
Abstract
Let be a field and be a square-free monomial ideal in the polynomial ring . The Green-Lazarsfeld index, , counts the number of steps to reach to a syzygy minimally generated by a nonlinear form in a graded minimal free resolution of . In this paper, we study this invariant for and its powers from a combinatorial point of view. We characterize all square-free monomial ideals generated in degree such that . Utilizing this result, we also characterize all square-free monomial ideals generated in degree such that and . In case , it is shown that for all if is any square-free monomial ideal with .
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Taxonomy
TopicsCommutative Algebra and Its Applications · Polynomial and algebraic computation · Algebraic Geometry and Number Theory
