nLab module over a C^∞-ring

Beck modules

The category of Beck modules over a C^∞-ring $A$ is equivalent to the category of ordinary modules over the underlying real algebra of $A$. A modern exposition of the proof can be found in Stel, Proposition 42 and in the more general context of Fermat theories in Carchedi–Roytenberg, Theorem 2.6.

The proof proceeds like the proof given at Beck module for ordinary rings, using the fact (Dubuc–Kock, Proposition 1.2) 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 as explained in Dubuc–Kock, the paragraph before Proposition 2.1.

Furthermore, by Dubuc–Kock [Proposition 2.2] the resulting notion of a Beck derivation coincides with that of a C^∞-derivation and by Dubuc–Kock [Theorem 2.3], the resulting module of Beck derivations can be computed as $I/I^2$, where $I$ is the kernel of the codiagonal $A\otimes A\to A$.

Kainz–Kriegl–Michor modules

A different, nonequivalent definition was proposed by Kainz–Kriegl–Michor in 1987.

Suppose $k$ is a commutative ring. Denote by $Poly_k$ the following category. Objects are $k$-modules. Morphisms $M\to N$ are polynomial maps $M\to N$, i.e., elements of $Sym M^*\otimes_k N$.

A commutative algebra $A$ can be identified with a product-preserving functor $FinPoly_k\to Set$, where $FinPoly_k$ is the full subcategory of $Poly_k$ on finitely generated free modules. The value $A(X)$ for $X\in FinPoly_k$ can be thought of as the space of regular functions? $Spec A\to X$, where $Spec A$ 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 $X\in CartPoly$)

$$\eta\colon Poly_k(X,M)\times A(X)\to M.$$

We require $\eta$ to be linear in the first argument.

That is to say, to specify an $A$-module $M$, we have to single out polynomial maps $k^n\to M$, together with a way to compose a polynomial map $k^n\to M$ with a regular function? $Spec A\to k^n$, obtaining a regular map $Spec A\to M$. Interpreting $M$ as the module of sections of a quasicoherent sheaf over $Spec A$, a regular map $Spec A\to M$ can be restricted to the diagonal $Spec A$, 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

$$\eta\colon C^\infty(X,M)\times A(X)\to M$$

that is linear in the first argument. If $\eta$ 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.

Related concepts

• Beck module, Beck derivation

References

• E. J. Dubuc, A. Kock, On 1-form classifiers, Communications in Algebra 12 12 (1984) 1471-1531 [doi:10.1080/00927878408823064]

• Herman Stel: $\infty$-Stacks and their Function Algebras with applications to $\infty$-Lie theory, MSc, University of Utrecht (2010) [pdf, webpage]

• David Carchedi, Dmitry Roytenberg, Homological Algebra for Superalgebras of Differentiable Functions, arXiv.

• G. Kainz, Andreas Kriegl, Peter Michor: $C^\infty$-algebras from the functional analytic view point, Journal of Pure and Applied Algebra 46 1 (1987) 89-107 [doi:10.1016/0022-4049(87)90045-4 pdf]