Hermitian categories, extension of scalars and systems of sesquilinear forms

Eva Bayer-Fluckiger, Uriya A. First and Daniel A. Moldovan

Vol. 270 (2014), No. 1, 1–26

DOI: 10.2140/pjm.2014.270.1

Abstract

We prove that the category of systems of sesquilinear forms over a given hermitian category is equivalent to the category of unimodular $1$-hermitian forms over another hermitian category. The sesquilinear forms are not required to be unimodular or defined on a reflexive object (i.e., the standard map from the object to its double dual is not assumed to be bijective), and the forms in the system can be defined with respect to different hermitian structures on the given category. This extends an earlier result of the first and third authors.

We use the equivalence to define a Witt group of sesquilinear forms over a hermitian category and to generalize results such as Witt’s cancellation theorem, Springer’s theorem, the weak Hasse principle, and finiteness of genus to systems of sesquilinear forms over hermitian categories.

Keywords

sesquilinear forms, hermitian forms, systems of sesquilinear forms, hermitian categories, additive categories, $K$-linear categories, scalar extension, Witt group

Mathematical Subject Classification 2010

Primary: 11E39, 11E81

Milestones

Received: 6 May 2013

Revised: 3 December 2013

Accepted: 10 January 2014

Published: 2 August 2014