Asymptotic v-numbers of graded (co)homology modules involving powers of an ideal

Let R be a Noetherian N-graded ring. Let L, M and N be finitely generated graded R-modules with N ⊆ M. For a homogeneous ideal I, and for each fixed k ∈ N, we show the asymptotic linearity of v-numbers of the graded modules ExtRᵏ(L,IⁿM/IⁿN) and TorₖR(L,IⁿM/IⁿN) as functions of n. Moreover, under some conditions on ExtRᵏ(L,M) and TorₖR(L,M) respectively, we prove similar behaviour for v-numbers of ExtRᵏ(L,M/IⁿN) and TorₖR(L,M/IⁿN). The last result is obtained by proving the asymptotic linearity of v-number of (U+IⁿV)/IⁿW, where U, V and W are graded submodules of a finitely generated graded R-module such that W ⊆ V and (0:UI) = 0.