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

Abstract:

Let $(A,\mathfrak {m})$ be a Cohen-Macaulay local ring and let $\mathrm {CM}(A)$ be the category of maximal Cohen-Macaulay $A$-modules. We construct $T \colon \mathrm {CM}(A)\times \mathrm {CM}(A) \rightarrow \operatorname {mod}(A)$, a subfunctor of $\operatorname {Ext}^1_A(-, -)$ and use it to study properties of associated graded modules over $G(A) = \bigoplus _{n\geq 0} \mathfrak {m}^n/\mathfrak {m}^{n+1}$, 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 $M$).