Pfister's local-global principle for Azumaya algebras with involution

TL;DR

This paper proves Pfister's local-global principle for hermitian forms over Azumaya algebras with involution, demonstrating that the Witt group is 2-primary torsion, using a hermitian Sylvester's law of inertia.

Contribution

It establishes a hermitian version of Sylvester's law of inertia and applies it to prove Pfister's local-global principle in this context.

Findings

Witt group of nonsingular hermitian forms is 2-primary torsion

Hermitian Sylvester's law of inertia is developed

Connections between hermitian forms, signatures, and positive semidefinite quadratic forms are explored

Abstract

We prove Pfister's local-global principle for hermitian forms over Azumaya algebras with involution over semilocal rings, and show in particular that the Witt group of nonsingular hermitian forms is $2$-primary torsion. Our proof relies on a hermitian version of Sylvester's law of inertia, which is obtained from an investigation of the connections between a pairing of hermitian forms extensively studied by Garrel, signatures of hermitian forms, and positive semidefinite quadratic forms.