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The linear syzygy graph of a monomial ideal and linear resolutions

Erfan Manouchehri; Ali Soleyman Jahan

Czechoslovak Mathematical Journal (2021)

  • Volume: 71, Issue: 3, page 785-802
  • ISSN: 0011-4642

Abstract

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For each squarefree monomial ideal I ⊂ S = k [ x 1 , ... , 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 x u i = y u 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 Δ ≅ G I Δ ∨ is a cycle or a tree.

How to cite

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Manouchehri, Erfan, and Soleyman Jahan, Ali. "The linear syzygy graph of a monomial ideal and linear resolutions." Czechoslovak Mathematical Journal 71.3 (2021): 785-802. <http://eudml.org/doc/297484>.

@article{Manouchehri2021,
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.},
author = {Manouchehri, Erfan, Soleyman Jahan, Ali},
journal = {Czechoslovak Mathematical Journal},
keywords = {monomial ideal; linear resolution; linear quotient; variable-decomposability; Cohen-Macaulay simplicial complex},
language = {eng},
number = {3},
pages = {785-802},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The linear syzygy graph of a monomial ideal and linear resolutions},
url = {http://eudml.org/doc/297484},
volume = {71},
year = {2021},
}

TY - JOUR
AU - Manouchehri, Erfan
AU - Soleyman Jahan, Ali
TI - The linear syzygy graph of a monomial ideal and linear resolutions
JO - Czechoslovak Mathematical Journal
PY - 2021
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 71
IS - 3
SP - 785
EP - 802
AB - 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.
LA - eng
KW - monomial ideal; linear resolution; linear quotient; variable-decomposability; Cohen-Macaulay simplicial complex
UR - http://eudml.org/doc/297484
ER -

References

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