Asymptotic behaviour of Vasconcelos invariants for products and powers of graded ideals

[2401.17815] Asymptotic behaviour of Vasconcelos invariants for products and powers of graded ideals Skip to main content Search Submit Donate Log in Mathematics > Commutative Algebra arXiv:2401.17815 (math) [Submitted on 31 Jan 2024 ( v1 ), last revised 28 Aug 2026 (this version, v5)] Title: Asymptotic behaviour of Vasconcelos invariants for products and powers of graded ideals Authors: Luca Fiorindo , Dipankar Ghosh View PDF HTML (experimental) Abstract: Let $R$ be a commutative Noetherian $\mathbb{N}$-graded ring. Let $N\subseteq M$ be finitely generated $\mathbb{Z}$-graded $R$-modules. Let $I_1,\ldots,I_r$ be nonzero proper homogeneous ideals of $R$. Denote ${\bf I}^{\underline{n}}:=I_1^{n_1}\cdots I_r^{n_r}$ for $\underline{n}=(n_1,\dots,n_r)\in\mathbb{N}^r$. In this paper, we prove that the (local) Vasconcelos invariant of ${\bf I}^{\underline{n}}M/{\bf I}^{\underline{n}}N$ is eventually the minimum of finitely many linear functions in $\underline{n}$. The same holds for $M/{\bf I}^{\underline{n}}N$ under certain conditions. Some specific examples are provided, where these functions are not eventually linear in $\underline{n}$. However, when $R$ is a polynomial ring over a field, we show that the global Vasconcelos invariants of $R/{\bf I}^{\underline{n}}$ and ${\bf I}^{\underline{n}}/{\bf I}^{\underline{n}+\underline{1}}$ are, in fact, asymptotically linear in $\underline{n}$ with the leading coefficients given by the initial degrees of $I_1,\ldots,I_r$. The last result is surprising: It differs from the Castelnuovo-Mumford regularity, which is not always linear even over polynomial rings, as shown by Bruns-Conca. Comments: Revised version, Journal ref. Journal of Algebraic Combinatorics. Added Section 2, Remark 3.2, Paragraph 4.1 and Example 4.8. Using Lemma 2.10, we have elaborated the proof of the 2nd part of Theorem 3.1 Subjects: Commutative Algebra (math.AC) MSC classes: 13A02, 13A15, 13A30 Report number: JACO-D-25-00130R1 Cite as: arXiv:2401.17815 [math.AC] (or arXiv:2401.17815v5 [math.AC] for this version) https://doi.org/10.48550/arXiv.2401.17815 Focus to learn more arXiv-issued DOI via DataCite Journal reference: Journal of Algebraic Combinatorics 64, 43 (2026) Related DOI : https://doi.org/10.1007/s10801-026-01598-5 Focus to learn more DOI(s) linking to related resources Submission history From: Luca Fiorindo [ view email ] [v1] Wed, 31 Jan 2024 13:18:08 UTC (8 KB) [v2] Fri, 2 Feb 2024 17:10:28 UTC (8 KB) [v3] Sat, 29 Mar 2025 11:08:58 UTC (15 KB) [v4] Fri, 11 Apr 2025 10:57:29 UTC (18 KB) [v5] Fri, 28 Aug 2026 11:56:26 UTC (23 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.AC < prev | next > new | recent | 2024-01 Change to browse by: math References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

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