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

Abstract

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 $T:CM(A)\times CM(A)\to mod(A)$, a subfunctor of $Ex{t}_A^1(-,-)$ and use it to study properties of associated graded modules over $G(A)={⨁}_{n\ge 0}{\mathfrak{m}}^n/{\mathfrak{m}}^{n+1}$, 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 $M$).