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Special groups and quadratic forms over rings with non-zero-divisor coefficients

Volume 258 / 2022

M. Dickmann, F. Miraglia, H. Ribeiro Fundamenta Mathematicae 258 (2022), 153-209 MSC: Primary 11E81, 03C65, 18B35; Secondary 06E99, 16G30, 12D15, 46E25. DOI: 10.4064/fm137-12-2021 Published online: 14 April 2022

Abstract

We present an algebraic theory of diagonal quadratic forms with non-zero-divisor coefficients over preordered (commutative, unitary) rings $\langle A,T\rangle $, where $2$ is invertible and the preorder $T$ satisfies a mild additional requirement. We prove that several major results known to hold in the classical theory of quadratic forms over fields, like the Arason–Pfister Hauptsatz and Pfister’s local-global principle, carry over to any class of preordered rings satisfying a property of ${\mathfrak N}T$-quadratic faithfulness, a notion central to our results. We prove that this property holds, and hence the above-mentioned results are valid, for many classes of rings frequently met in practice, such as (i) reduced $f$-rings and some of their extensions, for which Marshall’s signature conjecture and a vast generalization of Sylvester’s inertia law are also true; and (ii) reduced partially ordered Noetherian rings and many of their quotients (a result of interest in real algebraic geometry). This paper provides a broad extension of the theory developed in [M. Dickmann and F. Miraglia, Mem. Amer. Math. Soc. 238 (2015), no. 1128] and of the methods employed therein.

Authors

  • M. DickmannInstitut de Mathématiques de Jussieu
    – Paris Rive Gauche
    CNRS, Sorbonne Université
    Université de Paris
    Bâtiment Sophie Germain
    Place Aurélie Nemours
    75013 Paris, France
    e-mail
  • F. MiragliaInstitute of Mathematics and Statistics
    University of São Paulo
    Rua do Matão, 1010, CEP 05508-090
    São Paulo, S.P., Brazil
    e-mail
  • H. RibeiroInstitute of Mathematics and Statistics
    University of São Paulo
    Rua do Matão, 1010, CEP 05508-090
    São Paulo, S.P., Brazil
    e-mail

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