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Faithfully quadratic rings

About this Title

M. Dickmann, Projets Logique Mathématique, and Topologie et Géométrie Algébriques, Institut de Mathématiques de Jussieu $-$ Paris Rive Gauche, Universités Paris 6 et 7, Paris, France and F. Miraglia, Departamento de Matemática, Instituto de Matemática e Estatística, Universidade de São Paulo, São Paulo, Brazil

Publication: Memoirs of the American Mathematical Society
Publication Year: 2015; Volume 238, Number 1128
ISBNs: 978-1-4704-1468-9 (print); 978-1-4704-2629-3 (online)
DOI: https://doi.org/10.1090/memo/1128
Published electronically: May 19, 2015
Keywords: Algebraic theory of quadratic forms, preordered rings, special groups, $f$-rings, algebras of continuous real-valued functions, Archimedean preorders with bounded inversion, $K$-theory of rings, real spectra of rings
MSC: Primary 11E81, 11E70, 12D15, 03C65, 06E99, 46E25, 54C40

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Table of Contents

Chapters

  • Preface
  • 1. Basic Concepts
  • 2. Rings and Special Groups
  • 3. The Notion of T-Faithfully Quadratic Ring. Some Basic Consequences
  • 4. Idempotents, Products and T-isometry
  • 5. First-Order Axioms for Quadratic Faithfulness
  • 6. Rings with Many Units
  • 7. Transversality of Representation in p-rings with Bounded Inversion
  • 8. Reduced $f$ -Rings
  • 9. Strictly Representable Rings
  • 10. Quadratic Form Theory over Faithfully Quadratic Rings

Abstract

In this monograph we extend the classical algebraic theory of quadratic forms over fields to diagonal quadratic forms with invertible entries over broad classes of commutative, unitary rings where $-1$ is not a sum of squares and $2$ is invertible. We accomplish this by:

(1) Extending the classical notion of matrix isometry of forms to a suitable notion of $T$-isometry, where $T$ is a preorder of the given ring, $A$, or $T = A^2$.

(2) Introducing in this context three axioms expressing simple properties of (value) representation of elements of the ring by quadratic forms, well-known to hold in the field case.

Under these axioms we prove that the ring-theoretic approach based on $T$-isometry coincides with the formal approach formulated in terms of reduced special groups. This guarantees, for rings verifying these axioms, the validity of a number of important structural properties, notably the Arason-Pfister Hauptsatz, Milnor’s mod 2 Witt ring conjecture, Marshall’s signature conjecture, uniform upper bounds for the Pfister index of quadratic forms, a local-global Sylvester inertia law, etc. We call ($T$)-faithfully quadratic rings verifying these axioms.

A significant part of the monograph is devoted to prove quadratic faithfulness of certain outstanding (classes of) rings; among them, rings with many units satisfying a mild additional requirement, reduced f -rings (herein rings of continuous real-valued functions), and strictly representable rings.

Obviously, $T$-quadratic faithfulness depends on both the ring and the preorder $T$. We isolate a property of preorders defined solely in terms of the real spectrum of a given ring—that we baptise unit-reflecting preorders— which, for an extensive class of preordered rings, $\langle A,T \rangle$, turns out to be equivalent to the $T$-quadratic faithfulness of $A$. We show, e.g., that all preorders on the ring of continuous real-valued functions on a compact Hausdorff are unit-reflecting; we also give examples where this property fails.

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References

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