Signatures of hermitian forms and "prime ideals" of Witt groups

TL;DR

This paper advances the understanding of hermitian form signatures and prime ideals in Witt groups, showing invariance under Morita equivalence and establishing a correspondence with integer morphisms.

Contribution

It demonstrates that a single reference form can replace a tuple in defining H-signatures and studies prime ideals of Witt groups, extending classical classifications.

Findings

H-signatures are invariant under Morita equivalence.

A single form can replace a tuple of reference forms.

H-signatures correspond to morphisms into the integers.

Abstract

In this paper a further study is made of $H$-signatures of hermitian forms, introduced previously by the authors. It is shown that a tuple of reference forms $H$ may be replaced by a single form and that the $H$-signature is invariant under Morita equivalence of algebras with involution. The "prime ideals" of the Witt group are studied, obtaining results that are analogues of the classification of prime ideals of the Witt ring by Harrison and Lorenz-Leicht. It follows that $H$-signatures canonically correspond to morphisms into the integers.