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1. • CommentRowNumber1.
   • CommentAuthorDmitri Pavlov
   • CommentTime1 day ago
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   Author: Dmitri Pavlov
   Format: MarkdownItexCreated: ## 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 $A$. 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]]. ## 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 * G. Kainz, A. Kriegl, P. Michor, _C∞-algebras from the functional analytic view point_, Journal of Pure and Applied Algebra 46:1 (1987), 89-107. [doi][KKM] [KKM]: https://doi.org/10.1016/0022-4049(87)90045-4 <a href="https://ncatlab.org/nlab/revision/modules+over+C%5E%E2%88%9E-rings/1">v1</a>, <a href="https://ncatlab.org/nlab/show/modules+over+C%5E%E2%88%9E-rings">current</a>

   Created:

   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 $A$.

   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.

   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

   • G. Kainz, A. Kriegl, P. Michor, C∞-algebras from the functional analytic view point, Journal of Pure and Applied Algebra 46:1 (1987), 89-107. doi

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorDmitri Pavlov
   • CommentTime1 day ago
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexSingularized the title. <a href="https://ncatlab.org/nlab/revision/module+over+a+C%5E%E2%88%9E-ring/1">v1</a>, <a href="https://ncatlab.org/nlab/show/module+over+a+C%5E%E2%88%9E-ring">current</a>

   Singularized the title.

   v1, current

3. • CommentRowNumber3.
   • CommentAuthorDmitri Pavlov
   • CommentTime1 day ago
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexAdded references. <a href="https://ncatlab.org/nlab/revision/diff/module+over+a+C%5E%E2%88%9E-ring/2">diff</a>, <a href="https://ncatlab.org/nlab/revision/module+over+a+C%5E%E2%88%9E-ring/2">v2</a>, <a href="https://ncatlab.org/nlab/show/module+over+a+C%5E%E2%88%9E-ring">current</a>

   Added references.

   diff, v2, current

4. • CommentRowNumber4.
   • CommentAuthorDmitri Pavlov
   • CommentTime1 day ago
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexAdded a reference to Stel's thesis. <a href="https://ncatlab.org/nlab/revision/diff/module+over+a+C%5E%E2%88%9E-ring/4">diff</a>, <a href="https://ncatlab.org/nlab/revision/module+over+a+C%5E%E2%88%9E-ring/4">v4</a>, <a href="https://ncatlab.org/nlab/show/module+over+a+C%5E%E2%88%9E-ring">current</a>

   Added a reference to Stel’s thesis.

   diff, v4, current