A Deligne pairing for Hermitian Azumaya modules

TL;DR

This paper introduces a generalized Deligne pairing for modules over Azumaya algebras on arithmetic surfaces, utilizing Hermitian metrics and determinant of cohomology to extend classical concepts.

Contribution

It defines a new Deligne pairing for Azumaya modules with Hermitian metrics, expanding the framework of pairings in arithmetic geometry.

Findings

Defined Hermitian metrics on Azumaya algebras and modules

Introduced a determinant of cohomology for Azumaya modules

Established properties of the generalized Deligne pairing

Abstract

In this short note we want to give a definition of a generalized Deligne pairing for modules over an Azumaya algebra on an arithmetic surface $X$. We do this by defining Hermitian metrics on the Azumaya algebra and on the modules in question. Then we go on and define the determinant of the cohomology for a pair of modules over an Azumaya algebra. Using this we give the definition of a generalized Deligne pairing and study some of its properties.