Cartan-Dieudonné Theorem for ${\mathcal {A}}$-Modules

Abstract

Like the classical Cartan-Dieudonné theorem, the sheaf-theoretic version shows that ${\mathcal {A}}$-isometries on a convenient ${\mathcal {A}}$-module ${\mathcal {E}}$ of rank n can be decomposed in at most n orthogonal symmetries (reflections) with respect to non-isotropic hyperplanes. However, the coefficient sheaf of ${\mathbb {C}}$-algebras ${\mathcal {A}}$ is assumed to be a PID ${\mathbb {C}}$-algebra sheaf and, if ${(\mathcal {E},\phi)}$ is a pairing with ${\phi}$ a non-degenerate ${\mathcal {A}}$-bilinear morphism, we assume that ${\mathcal {E}}$ has nowhere-zero (local) isotropic sections; but, for Riemannian sheaves of ${\mathcal {A}}$-modules, this is not necessarily required.