Continuity of total signature maps for Azumaya algebras with involution
Vincent Astier, Thomas Unger
TL;DR
The paper develops a framework to define and analyze total signatures of hermitian forms over Azumaya algebras with involution across the real spectrum $\,Sper\,R$. By extending Morita-based signatures and introducing reference forms (the $ ta$-signature), it secures canonical, continuous total signature maps on $\,Sper\,R$, even in the presence of nontrivial Azumaya structure. A Garrel-inspired product on hermitian forms yields a near multiplicativity for signatures, controlled by the index invariant $_\ullet$, and the authors prove that the set of nil-orderings is clopen. The work also provides constructive methods to obtain reference forms, analyzes the continuity properties under both constructible and Harrison topologies, and discusses limitations of Harrison-continuity in general. Overall, the results establish a robust foundation for continuity properties of total signatures in the Azumaya setting and pave the way for inertia-type principles and local-global results beyond central simple 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.