Continuity of total signature maps for Azumaya algebras with involution

TL;DR

This paper extends the theory of signatures of hermitian forms from central simple algebras to Azumaya algebras with involution, establishing a continuous signature map over the real spectrum of the base ring.

Contribution

It generalizes previous methods to Azumaya algebras, enabling the natural and continuous assignment of signatures of hermitian forms across the real spectrum.

Findings

Total signatures are continuous functions on the real spectrum.

The approach applies to Azumaya algebras with involution.

Extension of signature theory from central simple to Azumaya algebras.

Abstract

In this paper we continue our investigation of signatures of hermitian forms over Azumaya algebras with involution over commutative rings. We show that the approach used in an earlier paper for central simple algebras can be extended to Azumaya algebras and leads to a natural way of choosing the signature of a hermitian form at a given ordering, producing total signatures of hermitian forms that are continuous functions on the real spectrum of the base ring.