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Matijevic-Roberts type theorems, Rees rings and associated graded rings

Jun Horiuchi, Kazuma Shimomoto

TL;DR

The paper investigates how key ring-theoretic properties propagate among Noetherian rings, their Rees rings, extended Rees rings, and associated graded rings by formulating and linking three problems: lifting along extensions, ascent/descent along $G(I)$, and Matijevic–Roberts type phenomena. It develops a general local-to-global framework with two fundamental propositions that connect these problems, and applies the theory to seminormality and weak normality, obtaining MR-type results that transfer properties from $G(I)$ to $\mathscr R_+(I)$, $\mathscr R(I)$, and $A$. The work highlights the utility of Rees-type constructions in understanding singularities and their resolutions, and provides structural criteria for when graded-local conditions control global properties. Overall, the results offer a cohesive approach to deriving global ring properties from graded or localized data, with concrete applications to seminormal and weakly normal cases.

Abstract

The aim of this article is to investigate interrelated structures lying among three notable problems in commutative algebra. These are Lifting problem, Ascent/descent along associated graded rings, and Matijevic-Roberts type problem.

Matijevic-Roberts type theorems, Rees rings and associated graded rings

TL;DR

The paper investigates how key ring-theoretic properties propagate among Noetherian rings, their Rees rings, extended Rees rings, and associated graded rings by formulating and linking three problems: lifting along extensions, ascent/descent along , and Matijevic–Roberts type phenomena. It develops a general local-to-global framework with two fundamental propositions that connect these problems, and applies the theory to seminormality and weak normality, obtaining MR-type results that transfer properties from to , , and . The work highlights the utility of Rees-type constructions in understanding singularities and their resolutions, and provides structural criteria for when graded-local conditions control global properties. Overall, the results offer a cohesive approach to deriving global ring properties from graded or localized data, with concrete applications to seminormal and weakly normal cases.

Abstract

The aim of this article is to investigate interrelated structures lying among three notable problems in commutative algebra. These are Lifting problem, Ascent/descent along associated graded rings, and Matijevic-Roberts type problem.
Paper Structure (3 sections, 7 theorems, 6 equations)

This paper contains 3 sections, 7 theorems, 6 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Lifting problem and associated graded rings

Key Result

Proposition 3.1

Fix a subcategory $\mathbf{D} \subseteq \mathbf{C}$. Suppose that $\mathcal{P}$ satisfies $(\bf{P2})$ in $\mathbf{D}$ and that $(\bf{Gr})$ is solved for an ideal $I=yA$ for a nonzero divisor $y \in \frak{m}$ for a local ring $(A,\frak{m})$ in $\mathbf{D}$. Then $(\bf{Lift})$ is solved in $\mathbf{D}

Theorems & Definitions (14)

  • Definition 2.1
  • Proposition 3.1
  • proof
  • Theorem 3.2: Seydi
  • Proposition 3.3
  • proof
  • Proposition 3.5
  • proof
  • Theorem 3.6
  • proof
  • ...and 4 more