Let
R
R
be a finitely generated
N
\mathbb N
-graded algebra domain over a Noetherian ring and let
I
I
be a homogeneous ideal of
R
R
. Given
P
∈
A
s
s
(
R
/
I
)
P\in Ass(R/I)
one defines the
v
v
-invariant
v
P
(
I
)
v_P(I)
of
I
I
at
P
P
as the least
c
∈
N
c\in \mathbb N
such that
P
=
I
:
f
P=I:f
for some
f
∈
R
c
f\in R_c
. A classical result of Brodmann [Proc. Amer. Math. Soc. 74 (1979), pp. 16–18] asserts that
A
s
s
(
R
/
I
n
)
Ass(R/I^n)
is constant for large
n
n
. So it makes sense to consider a prime ideal
P
∈
A
s
s
(
R
/
I
n
)
P\in Ass(R/I^n)
for all the large
n
n
and investigate how
v
P
(
I
n
)
v_P(I^n)
depends on
n
n
. We prove that
v
P
(
I
n
)
v_P(I^n)
is eventually a linear function of
n
n
. When
R
R
is the polynomial ring over a field this statement has been proved independently also by Ficarra and Sgroi in their recent preprint [Asymptotic behaviour of the
v
\text {v}
-number of homogeneous ideals, https://arxiv.org/abs/2306.14243, 2023].