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ON THE ASYMPTOTIC BEHAVIOR OF THE VASCONCELOS INVARIANT FOR GRADED MODULES
- Part of
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- Journal:
- Nagoya Mathematical Journal , First View
- Published online by Cambridge University Press:
- 10 February 2025, pp. 1-15
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- Article
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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.
