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arXiv · 0804.3481

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

A. Mallios·PP Ntumba
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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 $Φ: \mathcal{F}\times \mathcal{E}\longrightarrow \mathcal{A}$ induces a \textit{non-degenerate} bilinear $\mathcal{A}$-morphism $\barΦ: \mathcal{F}/\mathcal{E}^\perp\times \mathcal{E}/\mathcal{F}^\perp\longrightarrow \mathcal{A}$, 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.

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A. Mallios, PP Ntumba. 2008-04-22. Pairings of Sheaves of $\mathcal{A}$-Modules through Bilinear $\mathcal{A}$-Morphisms. https://arxiv.org/abs/0804.3481

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