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SIGNATURES OF HERMITIAN FORMS AND THE KNEBUSCH TRACE VINCENT ASTIER AND THOMAS UNGER
 

Summary: SIGNATURES OF HERMITIAN FORMS AND THE KNEBUSCH TRACE
FORMULA
VINCENT ASTIER AND THOMAS UNGER
Abstract. Signatures of quadratic forms have been generalized to hermitian forms
over algebras with involution. In the literature this is done via Morita theory, which
causes sign ambiguities in certain cases. In this paper, a hermitian version of the
Knebusch Trace Formula is established and used as a main tool to resolve these am-
biguities.
1. Introduction
In this paper we study signatures of hermitian forms over central simple algebras
with involution of any kind, defined over formally real fields. These generalize the
classical signatures of quadratic forms.
Following [3] we do this via extension to real closures and Morita equivalence. This
leads to the notion of M-signature of hermitian forms in Section 3.2. We study its
properties, make a detailed analysis of the impact of choosing different real closures
and different Morita equivalences and show in particular that sign changes can occur.
This motivates the search for a more intrinsic notion of signature, where such sign
changes do not occur.
In Section 3.3 we define such a signature, the H-signature, which only depends on
the choice of a tuple of hermitian forms, mimicking the fact that in quadratic form

  

Source: Astier, Vincent - Department of Mathematics, University College Dublin

 

Collections: Mathematics