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symplectic $\mathcal A$-modules; symplectic Gram-Schmidt theorem; symplectic basis; orthosymmetric $\mathcal {A}$-bilinear forms; orthogonal/symplectic geometry; strict integral domain algebra sheaf
Like the classical Gram-Schmidt theorem for symplectic vector spaces, the sheaf-theoretic version (in which the coefficient algebra sheaf $\mathcal A$ is appropriately chosen) shows that symplectic $\mathcal A$-morphisms on free $\mathcal A$-modules of finite rank, defined on a topological space $X$, induce canonical bases (Theorem 1.1), called symplectic bases. Moreover (Theorem 2.1), if $(\mathcal {E}, \phi )$ is an $\mathcal A$-module (with respect to a $\mathbb C$-algebra sheaf $\mathcal A$ without zero divisors) equipped with an orthosymmetric $\mathcal A$-morphism, we show, like in the classical situation, that “componentwise” $\phi $ is either symmetric (the (local) geometry is orthogonal) or skew-symmetric (the (local) geometry is symplectic). Theorem 2.1 reduces to the classical case for any free $\mathcal A$-module of finite rank.
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