Stanley depth of monomial ideals with small number of generators
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arXiv
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| Formato: | Preprint |
| Publicado: |
2009
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| _version_ | 1866917652883046400 |
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| author | Cimpoeas, Mircea |
| author_facet | Cimpoeas, Mircea |
| contents | For a monomial ideal $I\subset S=K[x_1,...,x_n]$, we show that $\sdepth(S/I)\geq n-g(I)$, where $g(I)$ is the number of the minimal monomial generators of $I$. If $I=vI'$, where $v\in S$ is a monomial, then we see that $\sdepth(S/I)=\sdepth(S/I')$. We prove that if $I$ is a monomial ideal $I\subset S$ minimally generated by three monomials, then $I$ and $S/I$ satisfy the Stanley conjecture. Given a saturated monomial ideal $I\subset K[x_1,x_2,x_3]$ we show that $\sdepth(I)=2$. As a consequence, $\sdepth(I)\geq \sdepth(K[x_1,x_2,x_3]/I)+1$ for any monomial ideal in $I\subset K[x_1,x_2,x_3]$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_0906_1105 |
| institution | arXiv |
| publishDate | 2009 |
| record_format | arxiv |
| spellingShingle | Stanley depth of monomial ideals with small number of generators Cimpoeas, Mircea Commutative Algebra 13H10, 13P10 For a monomial ideal $I\subset S=K[x_1,...,x_n]$, we show that $\sdepth(S/I)\geq n-g(I)$, where $g(I)$ is the number of the minimal monomial generators of $I$. If $I=vI'$, where $v\in S$ is a monomial, then we see that $\sdepth(S/I)=\sdepth(S/I')$. We prove that if $I$ is a monomial ideal $I\subset S$ minimally generated by three monomials, then $I$ and $S/I$ satisfy the Stanley conjecture. Given a saturated monomial ideal $I\subset K[x_1,x_2,x_3]$ we show that $\sdepth(I)=2$. As a consequence, $\sdepth(I)\geq \sdepth(K[x_1,x_2,x_3]/I)+1$ for any monomial ideal in $I\subset K[x_1,x_2,x_3]$. |
| title | Stanley depth of monomial ideals with small number of generators |
| topic | Commutative Algebra 13H10, 13P10 |
| url | https://arxiv.org/abs/0906.1105 |