| Title:
|
The linear syzygy graph of a monomial ideal and linear resolutions (English) |
| Author:
|
Manouchehri, Erfan |
| Author:
|
Soleyman Jahan, Ali |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
71 |
| Issue:
|
3 |
| Year:
|
2021 |
| Pages:
|
785-802 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
monomial ideal |
| Keyword:
|
linear resolution, linear quotient |
| Keyword:
|
variable-decomposability |
| Keyword:
|
Cohen-Macaulay simplicial complex |
| MSC:
|
13D02 |
| MSC:
|
13F20 |
| MSC:
|
13F55 |
| idZBL:
|
07396197 |
| idMR:
|
MR4295245 |
| DOI:
|
10.21136/CMJ.2020.0099-20 |
| . |
| Date available:
|
2021-08-02T08:06:35Z |
| Last updated:
|
2023-10-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/149056 |
| . |
| Reference:
|
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