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 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(InM) and v(InM/In+1M) are eventually linear functions of n. In addition, if (0:MI)=0, then v(M/InM) is also eventually linear with the same leading coefficient as that of v(InM/In+1M). These leading coefficients are described explicitly. The result on the linearity of v(M/InM) 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.