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It is proved that for any free $\mathcal{A}$-modules $\mathcal{F}$ and$\mathcal{E}$ of finite rank on some $\mathbb{C}$-algebraized space $(X,\mathcal{A})$ a \textit{degenerate} bilinear $\mathcal{A}$-morphism $\Phi:\mathcal{F}\times \mathcal{E}\longrightarrow \mathcal{A}$ induces a\textit{non-degenerate} bilinear $\mathcal{A}$-morphism $\bar{\Phi}:\mathcal{F}/\mathcal{E}^\perp\times\mathcal{E}/\mathcal{F}^\perp\longrightarrow \mathcal{A}$, where$\mathcal{E}^\perp$ and $\mathcal{F}^\perp$ are the \textit{orthogonal}sub-$\mathcal{A}$-modules associated with $\mathcal{E}$ and $\mathcal{F}$,respectively. This result generalizes the finite case of the classical result,which states that given two vector spaces $W$ and $V$, paired into a field $k$,the induced vector spaces $W/V^\perp$ and $V/W^\perp$ have the same dimension.Some related results are discussed as well.

Pairings of Sheaves of $\mathcal{A}$-Modules through Bilinear $\mathcal{A}$-Morphisms