A note on the $v$-invariant

Let $R$ be a finitely generated $\mathbb N$-graded algebra domain over a Noetherian ring and let $I$ be a homogeneous ideal of $R$. Given $P\in Ass(R/I)$ one defines the $v$-invariant $v_P(I)$ of $I$ at $P$ as the least $c\in \mathbb N$ such that $P=I:f$ for some $f\in R_c$. A classical result of Brodmann asserts that $Ass(R/I^n)$ is constant for large $n$. So it makes sense to consider a prime ideal $P\in Ass(R/I^n)$ for all the large $n$ and investigate how $v_P(I^n)$ depends on $n$. We prove that $v_P(I^n)$ is eventually a linear function of $n$. When $R$ is the polynomial ring over a field this statement has been proved independently also by Ficarra and Sgroi in a recent preprint.