ON THE ASYMPTOTIC BEHAVIOR OF THE VASCONCELOS INVARIANT FOR GRADED MODULES

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.