On the asymptotic behaviour of the Vasconcelos invariant for graded modules

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 behaviour 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 $(0{:}_{R}x)$ is a prime ideal of $R$. For a homogeneous ideal $I$ of $R$, we mainly prove that $v(I^{n}M)$ and $v(I^{n}M\mathrm{/}{I}^{n+1}M)$ are eventually linear functions of $n$. In addition, if $(0{:}_{M}I)=0$, then $v(M\mathrm{/}{I}^{n}M)$ is also eventually linear with the same leading coefficient as that of $v(I^{n}M\mathrm{/}{I}^{n+1}M)$. These leading coefficients are described explicitly. The result on the linearity of $v(M\mathrm{/}{I}^{n}M)$ 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.