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Quasi-quadratic modules in pseudo-valuation domain

Masato Fujita, Masaru Kageyama

TL;DR

This work classifies quasi-quadratic modules over a pseudo-valuation domain $A$ by reducing them to residue-valuation data via a pseudo-angular component map. The authors construct explicit modules $M_g(\mathcal{M})$ and $\Psi(M,g)$ and prove that any $\mathcal{M}$ decomposes as $\mathcal{M}=\bigcup_{g\ge e}\Psi(M_g(\mathcal{M}),g)$, establishing a bijection $\sigma: \mathfrak X_A \to \mathfrak S_{F_0}^F$ with inverse $\rho$, thereby transferring the problem to a product of residue modules. They provide a comprehensive decomposition, closure properties, and a finite-generation criterion, including a detailed treatment of the char two case in the appendix. The results yield a concrete, computable framework for describing quasi-quadratic modules in pseudo-valuation contexts and connect to concrete classifications in Euclidean-field cases and explicit ring examples.

Abstract

We study quasi-quadratic modules in a pseudo-valuation domain $A$ whose strict units admit a square root. Let $\mathfrak X_R^N$ denote the set of quasi-quadratic modules in an $R$-module $N$, where $R$ is a commutative ring. It is known that there exists a unique overring $B$ of $A$ such that $B$ is a valuation ring with the valuation group $(G,\leq)$ and the maximal ideal of $B$ coincides with that of $A$. Let $F$ be the residue field of $B$. In the above setting, we found a one-to-one correspondence between $\mathfrak X_A^A$ and a subset of $\prod_{g \in G,g \geq e} \mathfrak X_{F_0}^F$.

Quasi-quadratic modules in pseudo-valuation domain

TL;DR

This work classifies quasi-quadratic modules over a pseudo-valuation domain by reducing them to residue-valuation data via a pseudo-angular component map. The authors construct explicit modules and and prove that any decomposes as , establishing a bijection with inverse , thereby transferring the problem to a product of residue modules. They provide a comprehensive decomposition, closure properties, and a finite-generation criterion, including a detailed treatment of the char two case in the appendix. The results yield a concrete, computable framework for describing quasi-quadratic modules in pseudo-valuation contexts and connect to concrete classifications in Euclidean-field cases and explicit ring examples.

Abstract

We study quasi-quadratic modules in a pseudo-valuation domain whose strict units admit a square root. Let denote the set of quasi-quadratic modules in an -module , where is a commutative ring. It is known that there exists a unique overring of such that is a valuation ring with the valuation group and the maximal ideal of coincides with that of . Let be the residue field of . In the above setting, we found a one-to-one correspondence between and a subset of .
Paper Structure (7 sections, 27 theorems, 30 equations, 1 figure)

This paper contains 7 sections, 27 theorems, 30 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Review of the previous results
  3. Quasi-quadratic modules in a pseudo-valuation domain
  4. Special cases.
  5. Examples.
  6. Finitely generated quasi-quadratic modules of the pseudo-valuation ring.
  7. Appendix: When $F_0$ is of characteristic two

Key Result

Lemma 2.2

Let $\operatorname{\hbox{\bf p.an}}:K^{\times} \rightarrow F^{\times}$ be a pseudo-angular component map. For any $a \in K^{\times}$, $g \in G$ and $k \in F^{\times}$, there exists $u \in K^{\times}$ such that $\operatorname{\hbox{\bf val}}(u)=g$ and $\operatorname{\hbox{\bf p.an}}(au^2)=\operatorna

Figures (1)

  • Figure 1: $M_{\text{fan, oo}}(\theta_1,\theta_2)$, $M_{\text{fan, oc}}(\theta_1,\theta_2)$, $M_{\text{fan, co}}(\theta_1,\theta_2)$ and $M_{\text{fan, cc}}(\theta_1,\theta_2)$.

Theorems & Definitions (63)

  • Definition 1.1
  • Example 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 3.1
  • ...and 53 more