Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the structure of finitely generated modules and the unmixed degrees

Nguyen Tu Cuong, Pham Hung Quy

Abstract

Let $(R, \frak m)$ be a homomorphic image of a Cohen-Macaulay local ring and $M$ a finitely generated $R$-module. We use the splitting of local cohomology to shed a new light on the structure of non-Cohen-Macaulay modules. Namely, we show that every finitely generated $R$-module $M$ is associated by a sequence of invariant modules. This modules sequence expresses the deviation of $M$ with the Cohen-Macaulay property. This result generalizes the unmixed theorem of Cohen-Macaulayness for any finitely generated $R$-module. As an application we construct a new extended degree in sense of Vasconcelos.

On the structure of finitely generated modules and the unmixed degrees

Abstract

Let be a homomorphic image of a Cohen-Macaulay local ring and a finitely generated -module. We use the splitting of local cohomology to shed a new light on the structure of non-Cohen-Macaulay modules. Namely, we show that every finitely generated -module is associated by a sequence of invariant modules. This modules sequence expresses the deviation of with the Cohen-Macaulay property. This result generalizes the unmixed theorem of Cohen-Macaulayness for any finitely generated -module. As an application we construct a new extended degree in sense of Vasconcelos.

Paper Structure

This paper contains 5 sections, 36 theorems, 138 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The splitting of local cohomology
  4. The Cohen-Macaulay deviated sequences
  5. The unmixed degrees

Key Result

Theorem 1.1

Let $I$ be an ideal of $R$ and $x$ a parameter element of $M$ contained in $\frak b(M)^3$. Then for all $i < d - \dim R/I - 1$ we have

Theorems & Definitions (84)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 2.2
  • Definition 2.3
  • Proposition 2.4: C95, Theorem 2.6 (ii)
  • Definition 2.5: Hu82GY86
  • Definition 2.6: CC07-1
  • Proposition 2.7
  • Remark 2.8
  • ...and 74 more