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

TL;DR

This paper introduces a new subfunctor of Ext^1 for Cohen-Macaulay modules over a local ring and uses it to analyze the properties of associated graded modules, providing examples with bounded multiplicity.

Contribution

It constructs a specific subfunctor of Ext^1 and applies it to study Cohen-Macaulay modules and their associated graded modules, especially with bounded multiplicity.

Findings

Existence of Cohen-Macaulay rings with Cohen-Macaulay associated graded rings

Examples of modules with bounded multiplicity and Cohen-Macaulay associated graded modules

New subfunctor of Ext^1 for Cohen-Macaulay modules

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 $T \colon CM(A)\times CM(A) \rightarrow mod(A)$, a subfunctor of $Ext^1_A(-, -)$ and use it to study properties of associated graded modules over $G(A) = \bigoplus_{n\geq 0} \mathfrak{m}^n/\mathfrak{m}^{n+1}$, 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 $M$).