In this paper we study intersection multiplicities of modules as defined by Serre and prove that over regular local rings of
dim
⩽
5
\dim \leqslant 5
, given two modules
M
,
N
M,N
with
l
(
M
⊗
R
N
)
>
∞
l(M\otimes _{R}N) > \infty
and
dim
M
+
dim
N
>
dim
R
,
χ
(
M
,
N
)
=
∑
i
=
0
dim
R
(
−
1
)
i
l
(
Tor
i
R
(
M
,
N
)
)
=
0
\dim \;M + \dim \;N > \dim \;R,\chi (M,N) = \sum \nolimits _{i = 0}^{\dim \; R}( - 1)^i l(\operatorname {Tor}_i^R(M,N)) = 0
. We also study multiplicity in a more general set up. Finally we extend Serre’s result from pairs of modules to pairs of finite free complexes whose homologies are killed by
I
n
,
J
n
{I^n},{J^n}
, respectively, for some
n
>
0
n > 0
, with
dim
R
/
I
+
dim
R
/
J
>
dim
R
\dim \,R/I + \dim \,R/J > \dim \,R
.