Special groups and quadratic forms over rings with non-zero-divisor coefficients
Volume 258 / 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.
