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On Coprimary Filtrations

Yao Li

Abstract

The coprimary filtration is a basic construction in commutative algebra. In this article, we prove the existence and uniqueness of coprimary filtration of modules (not necessarily finitely generated) over a Noetherian ring. Moreover, we also prove the existence and uniqueness of coprimary filtrations of coherent sheaves over a locally Noetherian scheme.

On Coprimary Filtrations

Abstract

The coprimary filtration is a basic construction in commutative algebra. In this article, we prove the existence and uniqueness of coprimary filtration of modules (not necessarily finitely generated) over a Noetherian ring. Moreover, we also prove the existence and uniqueness of coprimary filtrations of coherent sheaves over a locally Noetherian scheme.

Paper Structure

This paper contains 7 sections, 27 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Associated points
  4. Reminder on partially ordered sets
  5. Coprimary filtrations
  6. Equivalent coprimary filtrations
  7. Direct sum decomposition

Key Result

Theorem 1.1

Let $A$ be a Noetherian ring and $M$ be a non-zero $A$-module. Let $(\mathop{\mathrm{Ass}}\nolimits(\widetilde{M}),<)$ be a well-ordering extension of $\mathop{\mathrm{Ass}}\nolimits(\widetilde{M})$ in Example Eg: key. Then there exists a unique filtration of $A$-modules, $(M^t)_{t\in \mathop{\mathr

Theorems & Definitions (60)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • ...and 50 more