Hermitian Categories, Extension of Scalars and Systems of Sesquilinear Forms

TL;DR

This paper establishes an equivalence between systems of sesquilinear forms over hermitian categories and unimodular 1-hermitian forms over another hermitian category, extending prior results and enabling new algebraic applications.

Contribution

It generalizes the theory of sesquilinear forms over hermitian categories and introduces a Witt ring framework for these systems, broadening the scope of classical theorems.

Findings

Equivalence between systems of sesquilinear forms and unimodular 1-hermitian forms.

Definition of a Witt ring for sesquilinear forms over hermitian categories.

Generalization of classical algebraic theorems to systems of sesquilinear forms.

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 a result obtained by E. Bayer-Fluckiger and D. Moldovan. We use the equivalence to define a Witt ring of sesquilinear forms over a hermitian category, and also to generalize various results (e.g.: Witt's Cancelation Theorem, Springer's Theorem, the weak Hasse principle, finiteness of genus) to systems of sesquilinear forms over hermitian categories.