English

Abstract

for each squarefree monomial ideal I ⊂ S = k[x1, …, xn], we associate a simple finite graph GI by using the first linear syzygies of I. The nodes of GI are the generators of I, and two vertices ui and uj are adjacent if there exist variables x, y such that xui = yuj. In the cases, where GI 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 GI, 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.