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Keywords:
monomial ideal; linear resolution, linear quotient; variable-decomposability; Cohen-Macaulay simplicial complex
monomial ideal; linear resolution, linear quotient; variable-decomposability; Cohen-Macaulay simplicial complex
Summary:
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.
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