The order of dominance of a monomial ideal
Let \(S\) be a polynomial ring in \(n\) variables over a field, and consider a monomial ideal \(M=(m_1,\ldots,m_q)\) of \(S\). We introduce a new invariant, called order of dominance of \(S/M\), and denoted \(\operatorname{odom}(S/M)\), which has many similarities with the codimension of \(S/M\). We...
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| Дата: | 2023 |
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| Мова: | English |
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Lugansk National Taras Shevchenko University
2023
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| Назва журналу: | Algebra and Discrete Mathematics |
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oai:ojs.admjournal.luguniv.edu.ua:article-17552023-06-18T17:42:42Z The order of dominance of a monomial ideal Alesandroni, G. monomial ideal, codimension, projective dimension, Betti number 13D02 Let \(S\) be a polynomial ring in \(n\) variables over a field, and consider a monomial ideal \(M=(m_1,\ldots,m_q)\) of \(S\). We introduce a new invariant, called order of dominance of \(S/M\), and denoted \(\operatorname{odom}(S/M)\), which has many similarities with the codimension of \(S/M\). We use the order of dominance to characterize the class of Scarf ideals that are Cohen-Macaulay, and also to characterize when the Taylor resolution is minimal. In addition, we show that \(\operatorname{odom}(S/M)\) has the following properties:(i) \(\operatorname{codim}(S/M) \leq \operatorname{odom}(S/M)\leq \operatorname{pd}(S/M)\).(ii) \(\operatorname{pd}(S/M)=n\) if and only if \(\operatorname{odom}(S/M)=n\).(iii) \(\operatorname{pd}(S/M)=1\) if and only if \(\operatorname{odom}(S/M)=1\).(iv) If \(\operatorname{odom}(S/M)=n-1\), then \(\operatorname{pd}(S/M)=n-1\).(v) If \(\operatorname{odom}(S/M)=q-1\), then \(\operatorname{pd}(S/M)=q-1\).(vi) If \(n=3\), then \(\operatorname{pd}(S/M)=\operatorname{odom}(S/M)\). Lugansk National Taras Shevchenko University 2023-06-18 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1755 10.12958/adm1755 Algebra and Discrete Mathematics; Vol 35, No 1 (2023) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1755/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/1755/818 Copyright (c) 2023 Algebra and Discrete Mathematics |
| institution |
Algebra and Discrete Mathematics |
| collection |
OJS |
| language |
English |
| topic |
monomial ideal codimension projective dimension Betti number 13D02 |
| spellingShingle |
monomial ideal codimension projective dimension Betti number 13D02 Alesandroni, G. The order of dominance of a monomial ideal |
| topic_facet |
monomial ideal codimension projective dimension Betti number 13D02 |
| format |
Article |
| author |
Alesandroni, G. |
| author_facet |
Alesandroni, G. |
| author_sort |
Alesandroni, G. |
| title |
The order of dominance of a monomial ideal |
| title_short |
The order of dominance of a monomial ideal |
| title_full |
The order of dominance of a monomial ideal |
| title_fullStr |
The order of dominance of a monomial ideal |
| title_full_unstemmed |
The order of dominance of a monomial ideal |
| title_sort |
order of dominance of a monomial ideal |
| description |
Let \(S\) be a polynomial ring in \(n\) variables over a field, and consider a monomial ideal \(M=(m_1,\ldots,m_q)\) of \(S\). We introduce a new invariant, called order of dominance of \(S/M\), and denoted \(\operatorname{odom}(S/M)\), which has many similarities with the codimension of \(S/M\). We use the order of dominance to characterize the class of Scarf ideals that are Cohen-Macaulay, and also to characterize when the Taylor resolution is minimal. In addition, we show that \(\operatorname{odom}(S/M)\) has the following properties:(i) \(\operatorname{codim}(S/M) \leq \operatorname{odom}(S/M)\leq \operatorname{pd}(S/M)\).(ii) \(\operatorname{pd}(S/M)=n\) if and only if \(\operatorname{odom}(S/M)=n\).(iii) \(\operatorname{pd}(S/M)=1\) if and only if \(\operatorname{odom}(S/M)=1\).(iv) If \(\operatorname{odom}(S/M)=n-1\), then \(\operatorname{pd}(S/M)=n-1\).(v) If \(\operatorname{odom}(S/M)=q-1\), then \(\operatorname{pd}(S/M)=q-1\).(vi) If \(n=3\), then \(\operatorname{pd}(S/M)=\operatorname{odom}(S/M)\). |
| publisher |
Lugansk National Taras Shevchenko University |
| publishDate |
2023 |
| url |
https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1755 |
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