The linear syzygy graph of a monomial ideal and linear resolutions

Abstract

for each squarefree monomial ideal I ⊂ S = k[x₁, …, x_n], we associate a simple finite graph G_I by using the first linear syzygies of I. The nodes of G_I are the generators of I, and two vertices u_i and u_j are adjacent if there exist variables x, y such that xu_i = yu_j. In the cases, where G_I is a cycle or a tree, we show that I has a linear resolution if and only if I has linear quotients and if and only if I is variable-decomposable. In addition, with the same assumption on G_I, we characterize all squarefree monomial ideals with a linear resolution. Using our results, we characterize all Cohen-Macaulay codimension 2 monomial ideals with a linear resolution. As another application of our results, we also characterize all Cohen-Macaulay simplicial complexes in the case, where ${G_{\rm{\Delta }}} \cong {G_{{I_{\rm{\Delta }}} \vee }}$ is a cycle or a tree.