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Groups of invertible ideals of one-dimensional Prüfer domains as groups of integer-valued functions

Dario Spirito

Abstract

Let $G$ be a one-dimensional $\ell$-subgroup of the group $\mathcal{F}(X,\mathbb{Z})$ of integer-valued functions on a set $X$. We show that $G$ is free under some hypothesis on the spectrum of $G$ and on its quotient groups at the prime ideals. We translate this result in the context of the study of freeness of the group $\mathrm{Inv}(D)$ of invertible ideals of a Prüfer domain $D$: in particular, we introduce the class of \emph{dd-domains} as the class of Prüfer domains having a set $X$ that is dense in $\mathrm{Spec}(D)$ (with respect to the inverse topology) and whose localizations are DVRs. This class is exactly the class of Prüfer domains for which $\mathrm{Inv}(D)$ is isomorphic (as an $\ell$-group) to a subgroup of $\mathcal{F}(X,\mathbb{Z})$.

Groups of invertible ideals of one-dimensional Prüfer domains as groups of integer-valued functions

Abstract

Let be a one-dimensional -subgroup of the group of integer-valued functions on a set . We show that is free under some hypothesis on the spectrum of and on its quotient groups at the prime ideals. We translate this result in the context of the study of freeness of the group of invertible ideals of a Prüfer domain : in particular, we introduce the class of \emph{dd-domains} as the class of Prüfer domains having a set that is dense in (with respect to the inverse topology) and whose localizations are DVRs. This class is exactly the class of Prüfer domains for which is isomorphic (as an -group) to a subgroup of .
Paper Structure (16 sections, 32 theorems, 56 equations)

This paper contains 16 sections, 32 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and notation
  3. Prüfer domains
  4. Invertible ideals
  5. Free groups
  6. $\ell$-groups
  7. Prüfer domains and $\ell$-groups
  8. The inverse and constructible topologies
  9. Groups of functions
  10. Primes at infinity
  11. dd-domains
  12. Derived sets
  13. Basic limit groups
  14. The successor ordinal case
  15. The limit ordinal case
  16. ...and 1 more sections

Key Result

Theorem 2.1

fuchs-abeliangroups Let $G$ be a group and $\{H_\alpha\}_{\alpha<\lambda}$ be a smooth sequence of subgroups such that $H_0=0$ and $\bigcup_{\alpha<\lambda}H_\alpha=G$. If $H_{\alpha+1}/H_\alpha$ is free for every $\alpha$, then $G$ is free.

Theorems & Definitions (75)

  • Theorem 2.1
  • Remark 2.2
  • Proposition 3.1
  • proof
  • Definition 3.2
  • Remark 3.3
  • Lemma 3.4
  • Definition 4.1
  • Theorem 4.2
  • proof
  • ...and 65 more