On the group sheaf of $$\mathcal{A}$$-symplectomorphisms

Abstract This is a part of a further undertaking to affirm that most of classical module theory may be retrieved in the framework of Abstract Differential Geometry (à la Mallios). More precisely, within this article, we study some defining basic concepts of symplectic geometry on free $$\mathcal{A}$$-modules by focussing in particular on the group sheaf of $$\mathcal{A}$$-symplectomorphisms, where $$\mathcal{A}$$ is assumed to be a torsion-free PID ℂ-algebra sheaf. The main result arising hereby is that $$\mathcal{A}$$-symplectomorphisms locally are products of symplectic transvections, which is a particularly well-behaved counterpart of the classical result.