Asymptotic v-numbers of graded (co)homology modules involving powers of an ideal
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866929747999588352 |
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| author | Ghosh, Dipankar Pramanik, Siddhartha |
| author_facet | Ghosh, Dipankar Pramanik, Siddhartha |
| contents | Let $R$ be a Noetherian $\mathbb{N}$-graded ring. Let $L$, $M$ and $N$ be finitely generated graded $R$-modules with $N \subseteq M$. For a homogeneous ideal $I$, and for each fixed $k \in \mathbb{N}$, we show the asymptotic linearity of v-numbers of the graded modules $ {\rm Ext}_R^{k}(L,{I^{n}M}/{I^{n}N})$ and ${\rm Tor}_k^{R}(L,{I^{n}M}/{I^{n}N})$ as functions of $n$. Moreover, under some conditions on ${\rm Ext}_R^k(L,M)$ and ${\rm Tor}_k^R(L,M)$ respectively, we prove similar behaviour for v-numbers of ${\rm Ext}_R^{k}(L,{M}/{I^{n}N})$ and $ {\rm Tor}_k^{R}(L,{M}/{I^{n}N})$. The last result is obtained by proving the asymptotic linearity of v-number of $(U+I^{n}V)/I^{n}W$, where $U$, $V$ and $W$ are graded submodules of a finitely generated graded $R$-module such that $W \subseteq V$ and $(0:_{U}I) = 0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_19992 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Asymptotic v-numbers of graded (co)homology modules involving powers of an ideal Ghosh, Dipankar Pramanik, Siddhartha Commutative Algebra 13D07, 13A02, 13A15 Let $R$ be a Noetherian $\mathbb{N}$-graded ring. Let $L$, $M$ and $N$ be finitely generated graded $R$-modules with $N \subseteq M$. For a homogeneous ideal $I$, and for each fixed $k \in \mathbb{N}$, we show the asymptotic linearity of v-numbers of the graded modules $ {\rm Ext}_R^{k}(L,{I^{n}M}/{I^{n}N})$ and ${\rm Tor}_k^{R}(L,{I^{n}M}/{I^{n}N})$ as functions of $n$. Moreover, under some conditions on ${\rm Ext}_R^k(L,M)$ and ${\rm Tor}_k^R(L,M)$ respectively, we prove similar behaviour for v-numbers of ${\rm Ext}_R^{k}(L,{M}/{I^{n}N})$ and $ {\rm Tor}_k^{R}(L,{M}/{I^{n}N})$. The last result is obtained by proving the asymptotic linearity of v-number of $(U+I^{n}V)/I^{n}W$, where $U$, $V$ and $W$ are graded submodules of a finitely generated graded $R$-module such that $W \subseteq V$ and $(0:_{U}I) = 0$. |
| title | Asymptotic v-numbers of graded (co)homology modules involving powers of an ideal |
| topic | Commutative Algebra 13D07, 13A02, 13A15 |
| url | https://arxiv.org/abs/2405.19992 |