On the asymptotic behaviour of the Vasconcelos invariant for graded modules
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866909530586087424 |
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| author | Fiorindo, Luca Ghosh, Dipankar |
| author_facet | Fiorindo, Luca Ghosh, Dipankar |
| contents | The notion of Vasconcelos invariant, known in the literature as v-number, of a homogeneous ideal in a polynomial ring over a field was introduced in 2020 to study the asymptotic behaviour of the minimum distance of projective Reed-Muller type codes. We initiate the study of this invariant for graded modules. Let $R$ be a Noetherian $\mathbb{N}$-graded ring, and $M$ be a finitely generated graded $R$-module. The v-number $v(M)$ can be defined as the least possible degree of a homogeneous element $x$ of $M$ for which $(0:_Rx)$ is a prime ideal of $R$. For a homogeneous ideal $I$ of $R$, we mainly prove that $v(I^nM)$ and $v(I^nM/I^{n+1}M)$ are eventually linear functions of $n$. In addition, if $(0:_M I)=0$, then $v(M/I^{n}M)$ is also eventually linear with the same leading coefficient as that of $v(I^nM/I^{n+1}M)$. These leading coefficients are described explicitly. The result on the linearity of $v(M/I^{n}M)$ considerably strengthens a recent result of Conca which was shown when $R$ is a domain and $M=R$, and Ficarra-Sgroi where the polynomial case is treated. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_16358 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the asymptotic behaviour of the Vasconcelos invariant for graded modules Fiorindo, Luca Ghosh, Dipankar Commutative Algebra 13A02, 13A15, 13A30, 13D45 The notion of Vasconcelos invariant, known in the literature as v-number, of a homogeneous ideal in a polynomial ring over a field was introduced in 2020 to study the asymptotic behaviour of the minimum distance of projective Reed-Muller type codes. We initiate the study of this invariant for graded modules. Let $R$ be a Noetherian $\mathbb{N}$-graded ring, and $M$ be a finitely generated graded $R$-module. The v-number $v(M)$ can be defined as the least possible degree of a homogeneous element $x$ of $M$ for which $(0:_Rx)$ is a prime ideal of $R$. For a homogeneous ideal $I$ of $R$, we mainly prove that $v(I^nM)$ and $v(I^nM/I^{n+1}M)$ are eventually linear functions of $n$. In addition, if $(0:_M I)=0$, then $v(M/I^{n}M)$ is also eventually linear with the same leading coefficient as that of $v(I^nM/I^{n+1}M)$. These leading coefficients are described explicitly. The result on the linearity of $v(M/I^{n}M)$ considerably strengthens a recent result of Conca which was shown when $R$ is a domain and $M=R$, and Ficarra-Sgroi where the polynomial case is treated. |
| title | On the asymptotic behaviour of the Vasconcelos invariant for graded modules |
| topic | Commutative Algebra 13A02, 13A15, 13A30, 13D45 |
| url | https://arxiv.org/abs/2401.16358 |