  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.
