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ON THE ASYMPTOTIC BEHAVIOR OF THE VASCONCELOS INVARIANT FOR GRADED MODULES

Part of: Commutative algebra: Homological methods General commutative ring theory

Published online by Cambridge University Press:  10 February 2025

LUCA FIORINDO Open the ORCID record for LUCA FIORINDO [Opens in a new window]  and
DIPANKAR GHOSH Open the ORCID record for DIPANKAR GHOSH [Opens in a new window]
LUCA FIORINDO*
Affiliation:
Dipartimento di Matematica, Dipartimento di Eccellenza 2023-2027 Università di Genova Genova - 16146 Italy
DIPANKAR GHOSH
Affiliation:
Department of Mathematics Indian Institute of Technology Kharagpur Kharagpur - 721302, West Bengal India dipankar@maths.iitkgp.ac.in, dipug23@gmail.com
*
luca.fiorindo@edu.unige.it
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Abstract

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 behavior 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.

Keywords

Graded rings and modulesAssociate primesv-numbersCastelnuovo–Mumford regularity

MSC classification

Primary: 13A02: Graded rings 13A15: Ideals; multiplicative ideal theory 13A30: Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics 13D45: Local cohomology
Type
Article
Information
Nagoya Mathematical Journal , First View , pp. 1 - 15
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Foundation Nagoya Mathematical Journal

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