In commutative algebra the Hilbert–Samuel function, named after David Hilbert and Pierre Samuel, of a nonzero finitely generated module
M
over a commutative Noetherian local ring
A
and a primary ideal
I
of
A
is the map
χ
M
I
:
N
→
N
such that, for all
n
∈
N
,
χ
M
I
(
n
)
=
ℓ
(
M
/
I
n
M
)
where
ℓ
denotes the length over
A
. It is related to the Hilbert function of the associated graded module
gr
I
(
M
)
by the identity
χ
M
I
(
n
)
=
∑
i
=
0
n
H
(
gr
I
(
M
)
,
i
)
.
For sufficiently large
n
, it coincides with a polynomial function of degree equal to
dim
(
gr
I
(
M
)
)
.
For the ring of formal power series in two variables
k
[
[
x
,
y
]
]
taken as a module over itself and graded by the order and the ideal generated by the monomials x2 and y3 we have
χ
(
1
)
=
1
,
χ
(
2
)
=
3
,
χ
(
3
)
=
5
,
χ
(
4
)
=
6
and
χ
(
k
)
=
6
for
k
>
4.
Unlike the Hilbert function, the Hilbert–Samuel function is not additive on an exact sequence. However, it is still reasonably close to being additive, as a consequence of the Artin–Rees lemma. We denote by
P
I
,
M
the Hilbert-Samuel polynomial; i.e., it coincides with the Hilbert–Samuel function for large integers.
Proof: Tensoring the given exact sequence with
R
/
I
n
and computing the kernel we get the exact sequence:
0
→
(
I
n
M
∩
M
′
)
/
I
n
M
′
→
M
′
/
I
n
M
′
→
M
/
I
n
M
→
M
″
/
I
n
M
″
→
0
,
which gives us:
χ
M
I
(
n
−
1
)
=
χ
M
′
I
(
n
−
1
)
+
χ
M
″
I
(
n
−
1
)
−
ℓ
(
(
I
n
M
∩
M
′
)
/
I
n
M
′
)
.
The third term on the right can be estimated by Artin-Rees. Indeed, by the lemma, for large n and some k,
I
n
M
∩
M
′
=
I
n
−
k
(
(
I
k
M
)
∩
M
′
)
⊂
I
n
−
k
M
′
.
Thus,
ℓ
(
(
I
n
M
∩
M
′
)
/
I
n
M
′
)
≤
χ
M
′
I
(
n
−
1
)
−
χ
M
′
I
(
n
−
k
−
1
)
.
This gives the desired degree bound.
