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

Abstract

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 $\Phi :\mathcal{F}\times \mathcal{E}⟶\mathcal{A}$ induces a \textit{non-degenerate} bilinear $\mathcal{A}$-morphism $\bar{\Phi }:\mathcal{F}/{\mathcal{E}}^{\perp }\times \mathcal{E}/{\mathcal{F}}^{\perp }⟶\mathcal{A}$, 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.

23 pages