Comparison of stability indices of powers of graded ideals
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912385944518656 |
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| author | Ficarra, Antonino Sgroi, Emanuele |
| author_facet | Ficarra, Antonino Sgroi, Emanuele |
| contents | In this paper, we compare the index of ass-stability $\text{astab}(I)$ and the index of $\text{v}$-stability $\text{vstab}(I)$ of powers of a graded ideal $I$. We prove that $\text{astab}(I)=1\le\text{vstab}(I)$ for any graded ideal $I$ in a 2-dimensional polynomial ring, and that $\text{vstab}(I)$ can be any positive integer in this situation. Moreover, given any integers $a,b\ge1$, we construct a graded ideal $I$ in a $3(a+1)$-dimensional polynomial ring such that $(\text{astab}(I),\text{vstab}(I))=(a,b)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_15608 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Comparison of stability indices of powers of graded ideals Ficarra, Antonino Sgroi, Emanuele Commutative Algebra Combinatorics In this paper, we compare the index of ass-stability $\text{astab}(I)$ and the index of $\text{v}$-stability $\text{vstab}(I)$ of powers of a graded ideal $I$. We prove that $\text{astab}(I)=1\le\text{vstab}(I)$ for any graded ideal $I$ in a 2-dimensional polynomial ring, and that $\text{vstab}(I)$ can be any positive integer in this situation. Moreover, given any integers $a,b\ge1$, we construct a graded ideal $I$ in a $3(a+1)$-dimensional polynomial ring such that $(\text{astab}(I),\text{vstab}(I))=(a,b)$. |
| title | Comparison of stability indices of powers of graded ideals |
| topic | Commutative Algebra Combinatorics |
| url | https://arxiv.org/abs/2505.15608 |