Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Smooth sequences and overring operators

Dario Spirito

TL;DR

The paper introduces smooth sequences of overrings and an overrings operator framework to study the structure of the group $Inv(D)$ of invertible ideals in integral domains, especially Prüfer domains. By linking algebraic structure to the inverse topology on $Spec(D)^{inv}$ and Cantor-Bendixson-type derivations, it develops criteria that reduce freeness of $Inv(D)$ to the freeness of overrings and their kernels. The framework unifies prior results on SP-scattered, InvXD, and pre-Jaffard settings and offers a constructive method to build smooth chains by iterating overrings operators. Applications to one-dimensional Prüfer domains yield decompositions of $Inv(D)$ via Theta-based constructions and enable reduction steps that streamline the freeness analysis.

Abstract

We introduce smooth sequences of integral domains as well-ordered ascending chains that behave well at limit ordinals. Subsequently, we use this notion to give some conditions on the freeness of kernels of extension maps between groups of invertible ideals of Prüfer domains. We also define overring operators to construct smooth sequences in a recursive way.

Smooth sequences and overring operators

TL;DR

The paper introduces smooth sequences of overrings and an overrings operator framework to study the structure of the group of invertible ideals in integral domains, especially Prüfer domains. By linking algebraic structure to the inverse topology on and Cantor-Bendixson-type derivations, it develops criteria that reduce freeness of to the freeness of overrings and their kernels. The framework unifies prior results on SP-scattered, InvXD, and pre-Jaffard settings and offers a constructive method to build smooth chains by iterating overrings operators. Applications to one-dimensional Prüfer domains yield decompositions of via Theta-based constructions and enable reduction steps that streamline the freeness analysis.

Abstract

We introduce smooth sequences of integral domains as well-ordered ascending chains that behave well at limit ordinals. Subsequently, we use this notion to give some conditions on the freeness of kernels of extension maps between groups of invertible ideals of Prüfer domains. We also define overring operators to construct smooth sequences in a recursive way.

Paper Structure

This paper contains 4 sections, 11 theorems, 21 equations.

Table of Contents

  1. Introduction
  2. Smooth sequences
  3. Overring operators
  4. Groups of invertible ideals

Key Result

Proposition 2.3

Let $D$ be a Prüfer domain and let $\mathcal{D}:=\{D_\alpha\}_{\alpha<\lambda}$ be a well-ordered ascending chain of overrings of $D$. For every $\alpha$, let $X_\alpha:=X(D_\alpha)$ and $Y_\alpha:=Y(D_\alpha)$ (following the notation above). Then, the following are equivalent:

Theorems & Definitions (29)

  • Definition 2.1
  • Example 2.2
  • Proposition 2.3
  • proof
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Remark 3.4
  • Example 3.5
  • Proposition 4.1
  • ...and 19 more