Reflexivity of orthogonality in A-modules

Abstract

In this paper, as part of a project initiated by A. Mallios consisting of exploring new horizons for Abstract Differential Geometry (`a la Mallios), [5, 6, 7, 8], such as those related to the classical symplectic geometry, we show that essential results pertaining to biorthogonality in pairings of vector spaces do hold for biorthogonality in pairings of A-modules. We single out that orthogonality is reflexive for orthogonally convenient pairings of free A-modules of finite rank, governed by non-degenerate A-morphisms, and where A is a PID (Corollary 3.8). For the rank formula (Corollary 3.3), the algebra sheaf A is assumed to be a PID. The rank formula relates the rank of an A-morphism and the rank of the kernel (sheaf) of the same A-morphism with the rank of the source free A-module of the A-morphism concerned.