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

TL;DR

This paper generalizes classical finite-dimensional pairing results to sheaves of modules over a complex algebraized space, showing how degenerate bilinear morphisms induce non-degenerate pairings on quotient modules.

Contribution

It introduces a method to derive non-degenerate pairings from degenerate ones for sheaves of modules, extending classical finite-dimensional duality results to a sheaf-theoretic context.

Findings

Degenerate bilinear morphisms induce non-degenerate pairings on quotient modules.

Generalization of classical duality to sheaves of modules over complex spaces.

Discussion of related results in sheaf-theoretic duality.

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 $\Phi: \mathcal{F}\times \mathcal{E}\longrightarrow \mathcal{A}$ 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.