A Knebusch trace formula for Azumaya algebras with involution

TL;DR

This paper extends Knebusch's trace formula to hermitian forms over Azumaya algebras with involution, providing new insights into signatures, local-global principles, and stability indices in algebraic structures.

Contribution

It introduces a trace formula for signatures of hermitian forms over Azumaya algebras with involution, generalizing previous results on symmetric bilinear forms.

Findings

Established a trace formula for hermitian form signatures.

Derived an exact sequence for total signatures in semilocal rings.

Connected results to Pfister's local-global principle and stability index.

Abstract

We establish a trace formula for signatures of hermitian forms over Azumaya algebras with involution, extending Knebusch's work on symmetric bilinear forms over finite \'etale extensions of commutative base rings. As an application when the base ring is semilocal, we obtain an exact sequence for total signatures, related to Pfister's local-global principle and the notion of stability index.