Pairings of Sheaves of A-Modules

Abstract

It is proved that for any free A-modules F and ε of finite rank on some C-algebraized space (X,A) a degenerate A-bilinear morphism Φ : F × ε → A induces a non-degenerate A-bilinear morphism Φ : F/ε ^⊥ × ε/F ^⊥ → A, where ε ^⊥ and F ^⊥ are the orthogonal sub-A-modules associated with ε and 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 ^⊥ and V/W ^⊥ have the same dimension. Some related results are discussed as well.