A sub-functor for Ext and Cohen-Macaulay associated graded modules with bounded multiplicity

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