| Title:
|
The cleanness of (symbolic) powers of Stanley-Reisner ideals (English) |
| Author:
|
Bandari, Somayeh |
| Author:
|
Jahan, Ali Soleyman |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
67 |
| Issue:
|
3 |
| Year:
|
2017 |
| Pages:
|
767-778 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $\Delta $ be a pure simplicial complex on the vertex set $[n]=\{1,\ldots ,n\}$ and $I_\Delta $ its Stanley-Reisner ideal in the polynomial ring $S=K[x_1,\ldots ,x_n]$. We show that $\Delta $ is a matroid (complete intersection) if and only if $S/I_\Delta ^{(m)}$ ($S/I_\Delta ^m$) is clean for all $m\in \mathbb {N}$ and this is equivalent to saying that $S/I_\Delta ^{(m)}$ ($S/I_\Delta ^m$, respectively) is Cohen-Macaulay for all $m\in \mathbb {N}$. By this result, we show that there exists a monomial ideal $I$ with (pretty) cleanness property while $S/I^m$ or $S/I^{(m)}$ is not (pretty) clean for all integer $m\geq 3$. If $\dim (\Delta )=1$, we also prove that $S/I_\Delta ^{(2)}$ ($S/I_\Delta ^2$) is clean if and only if $S/I_\Delta ^{(2)}$ ($S/I_\Delta ^2$, respectively) is Cohen-Macaulay. (English) |
| Keyword:
|
clean |
| Keyword:
|
Cohen-Macaulay simplicial complex |
| Keyword:
|
complete intersection |
| Keyword:
|
matroid |
| Keyword:
|
symbolic power |
| MSC:
|
05E40 |
| MSC:
|
13F20 |
| MSC:
|
13F55 |
| idZBL:
|
Zbl 06770129 |
| idMR:
|
MR3697915 |
| DOI:
|
10.21136/CMJ.2017.0173-16 |
| . |
| Date available:
|
2017-09-01T12:24:28Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/146858 |
| . |
| Reference:
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