Biorthogonality in $\mathcal{A}$-Pairings and Hyperbolic Decomposition Theorem for $\mathcal{A}$-Modules

In this paper, as part of a project initiated by A. Mallios consisting of exploring new horizons for $\mathit{\text{Abstract}}\mathit{\text{Differential}}\mathit{\text{Geometry}}$ ($\grave{a}$ la Mallios), \cite{mallios1997, mallios, malliosvolume2, modern}, such as those related to the $\mathit{\text{classical}}\mathit{\text{symplectic}}\mathit{\text{geometry}}$, we show that results pertaining to biorthogonality in pairings of vector spaces do hold for biorthogonality in pairings of $\mathcal{A}$-modules. However, for the $\mathit{\text{dimension}}\mathit{\text{formula}}$ the algebra sheaf $\mathcal{A}$ is assumed to be a PID. The dimension formula relates the rank of an $\mathcal{A}$-morphism and the dimension of the kernel (sheaf) of the same $\mathcal{A}$-morphism with the dimension of the source free $\mathcal{A}$-module of the $\mathcal{A}$-morphism concerned. Also, in order to obtain an analog of the Witt’s hyperbolic decomposition theorem, $\mathcal{A}$ is assumed to be a PID while topological spaces on which $\mathcal{A}$-modules are defined are assumed $\mathit{\text{connected}}$.