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The category of Beck modules over a C^∞-ring A is equivalent to the category of ordinary modules over the underlying real algebra of A.
This is established using the proof given at Beck module for ordinary rings, using the fact that ideals of C^∞-rings coincide with ideals in the ordinary sense and the square zero extension construction used there can be promoted to a C^∞-ring using Taylor expansions.
Furthermore, the resulting notion of a Beck derivation coincides with that of a C^∞-derivation.
A different, nonequivalent definition was proposed by Kainz–Kriegl–Michor in 1987.
Suppose k is a commutative ring. Denote by Polyk the following category. Objects are k-modules. Morphisms M→N are polynomial maps M→N, i.e., elements of SymM*⊗kN.
A commutative algebra A can be identified with a product-preserving functor FinPolyk→Set, where FinPolyk is the full subcategory of Polyk on finitely generated free modules. The value A(X) for X∈FinPolyk can be thought of as the space of regular functions SpecA→X, where SpecA is the Zariski spectrum of A.
The starting observation is that a module M over a commutative k-algebra A can be identified with a dinatural transformation (dinatural in X∈CartPoly)
η:Polyk(X,M)×A(X)→M.We require η to be linear in the first argument.
That is to say, to specify an A-module M, we have to single out polynomial maps kn→M, together with a way to compose a polynomial map kn→M with a regular function SpecA→kn, obtaining a regular map SpecA→M. Interpreting M as the module of sections of a quasicoherent sheaf over SpecA, a regular map SpecA→M can be restricted to the diagonal SpecA, obtaining an element of M as required.
The proposal of Kainz–Kriegl–Michor is then to replace polynomial maps with smooth maps:
A C^∞-module over a C^∞-ring A is a Hausdorff locally convex topological vector space M together with a dinatural transformation
η:C∞(X,M)×A(X)→Mthat is linear in the first argument. If η is also continuous in the first argument, we say that M is a continuous C^∞-module.
Topological vector spaces in the above definition can be replaced by any notion of a vector space that allows for smooth maps, e.g., convenient vector space etc.
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