Let
(
A
,
m
)
(A,\mathfrak {m})
be a Cohen-Macaulay local ring and let
C
M
(
A
)
\mathrm {CM}(A)
be the category of maximal Cohen-Macaulay
A
A
-modules. We construct
T
:
C
M
(
A
)
×
C
M
(
A
)
→
mod
(
A
)
T \colon \mathrm {CM}(A)\times \mathrm {CM}(A) \rightarrow \operatorname {mod}(A)
, a subfunctor of
Ext
A
1
(
−
,
−
)
\operatorname {Ext}^1_A(-, -)
and use it to study properties of associated graded modules over
G
(
A
)
=
⨁
n
≥
0
m
n
/
m
n
+
1
G(A) = \bigoplus _{n\geq 0} \mathfrak {m}^n/\mathfrak {m}^{n+1}
, the associated graded ring of
A
A
. As an application we give several examples of complete Cohen-Macaulay local rings
A
A
with
G
(
A
)
G(A)
Cohen-Macaulay and having distinct indecomposable maximal Cohen-Macaulay modules
M
n
M_n
with
G
(
M
n
)
G(M_n)
Cohen-Macaulay and the set
{
e
(
M
n
)
}
\{e(M_n)\}
bounded (here
e
(
M
)
e(M)
denotes multiplicity of
M
M
).