Czechoslovak Mathematical Journal, Vol. 71, No. 3, pp. 785-802, 2021
The linear syzygy graph of a monomial ideal and linear resolutions
Erfan Manouchehri, Ali Soleyman Jahan
Received March 4, 2020. Published online November 18, 2020.
Abstract: For each squarefree monomial ideal $I\subset S = k[x_1,\ldots, 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 $G_{\Delta}\cong G_{I_{\Delta^{\vee}}}$ is a cycle or a tree.
Keywords: monomial ideal; linear resolution, linear quotient; variable-decomposability; Cohen-Macaulay simplicial complex
Affiliations: Erfan Manouchehri (corresponding author), Ali Soleyman Jahan, Department of Mathematics, University of Kurdistan, P.O.Box 66177-15175, Sanadaj, Iran, e-mail: erfanm6790@yahoo.com, solymanjahan@gmail.com
