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1. • CommentRowNumber1.
   • CommentAuthorDmitri Pavlov
   • CommentTimeSep 16th 2024
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexCreated: ## Idea In [[algebraic geometry]], the module of [[Kähler differentials]] of a [[commutative ring]] $R$ corresponds under the [[Serre–Swan duality]] to the [[cotangent bundle]] of the [[Zariski spectrum]] of $R$. In contrast, the module of [[Kähler differentials]] of the [[commutative real algebra]] of [[smooth functions]] on a [[smooth manifold]] $M$ receives a canonical map from the module of smooth sections of the [[cotangent bundle]] of $M$ that is quite far from being an isomorphism. An example illustrating this point is $M=\mathbf{R}$, since in the module of (traditionally defined) [[Kähler differentials]] of $C^\infty(M)$ we have $d(exp(x))\ne exp dx$, where $\exp\colon\mathbf{R}\to\mathbf{R}$ is the [[exponential function]]. That is to say, the traditional algebraic notion of a [[Kähler differential]] is unable to deduce that $\exp'=\exp$ using the Leibniz rule. However, this is not a defect in the conceptual idea itself, but merely a failure to use the correct formalism. The appropriate notion of a ring in the context of [[differential geometry]] is not merely a commutative real algebra, but a more refined structure, namely, a [[C^∞-ring]]. This notion comes with its own variant of [[commutative algebra]]. Some of the resulting concepts turn out to be exactly the same as in the traditional case. For example, ideals of [[C^∞-rings]] and [[modules]] over [[C^∞-rings]] happen to coincide with ideals and modules in the traditional sense. Others, like derivations, must be defined carefully, and definitions that used to be equivalent in the traditional algebraic context need not remain so in the context of [[C^∞-rings]]. Observe that a map of sets $d\colon A\to M$ (where $M$ is an $A$-module) is a derivation if and only if for any real [[polynomial]] $f(x_1,\ldots,x_n)$ the [[chain rule]] holds: $$d(f(a_1,\ldots,a_n))=\sum_i {\partial f\over\partial x_i}(x_1,\ldots,x_n) dx_i.$$ Indeed, taking $f(x_1,x_2)=x_1+x_2$ and $f(x_1,x_2)=x_1 x_2$ recovers the additivity and Leibniz property of derivations, respectively. Observe also that $f$ is an element of the free commutative real algebra on $n$ elements, i.e., $\mathbf{R}[x_1,\ldots,x_n]$. If we now substitute [[C^∞-rings]] for commutative real algebras, we arrive at the correct notion of a derivation for [[C^∞-rings]]: A __C^∞-derivation__ of a [[C^∞-ring]] $A$ is a map of sets $A\to M$ (where $M$ is a [[module]] over $A$) such that the following chain rule holds for every smooth function $f\in\mathrm{C}^\infty(\mathbf{R}^n)$: $$d(f(a_1,\ldots,a_n))=\sum_i {\partial f\over\partial x_i}(x_1,\ldots,x_n) dx_i,$$ where both sides use the structure of a [[C^∞-ring]] to evaluate a smooth real function on a collection of elements in $A$. The module of Kähler C^∞-differentials can now be defined in the same manner as ordinary [[Kähler differentials]], using C^∞-derivations instead of ordinary derivations. \begin{theorem} (Dubuc, Kock, 1984.) The module of Kähler C^∞-differentials of the [[C^∞-ring]] of smooth functions on a [[smooth manifold]] $M$ is canonically isomorphic to the module of sections of the [[cotangent bundle]] of $M$. \end{theorem} ## Related concepts * [[Kähler differential]], [[derivation]] * [[C^∞-ring]] ## References * [[E. J. Dubuc]], [[A. Kock]], _On 1-form classifiers_, Communications in Algebra 12:12 (1984), 1471–1531. [doi](https://doi.org/10.1080/00927878408823064). <a href="https://ncatlab.org/nlab/revision/K%C3%A4hler+C%5E%E2%88%9E-differentials+of+smooth+functions+are+differential+1-forms/1">v1</a>, <a href="https://ncatlab.org/nlab/show/K%C3%A4hler+C%5E%E2%88%9E-differentials+of+smooth+functions+are+differential+1-forms">current</a>

   Created:

   Idea

   In algebraic geometry, the module of Kähler differentials of a commutative ring $R$ corresponds under the Serre–Swan duality to the cotangent bundle of the Zariski spectrum of $R$.

   In contrast, the module of Kähler differentials of the commutative real algebra of smooth functions on a smooth manifold $M$ receives a canonical map from the module of smooth sections of the cotangent bundle of $M$ that is quite far from being an isomorphism.

   An example illustrating this point is $M=\mathbf{R}$, since in the module of (traditionally defined) Kähler differentials of $C^\infty(M)$ we have $d(exp(x))\ne exp dx$, where $\exp\colon\mathbf{R}\to\mathbf{R}$ is the exponential function. That is to say, the traditional algebraic notion of a Kähler differential is unable to deduce that $\exp'=\exp$ using the Leibniz rule.

   However, this is not a defect in the conceptual idea itself, but merely a failure to use the correct formalism. The appropriate notion of a ring in the context of differential geometry is not merely a commutative real algebra, but a more refined structure, namely, a C^∞-ring.

   This notion comes with its own variant of commutative algebra. Some of the resulting concepts turn out to be exactly the same as in the traditional case. For example, ideals of C^∞-rings and modules over C^∞-rings happen to coincide with ideals and modules in the traditional sense. Others, like derivations, must be defined carefully, and definitions that used to be equivalent in the traditional algebraic context need not remain so in the context of C^∞-rings.

   Observe that a map of sets $d\colon A\to M$ (where $M$ is an $A$-module) is a derivation if and only if for any real polynomial $f(x_1,\ldots,x_n)$ the chain rule holds:

   $$d(f(a_1,\ldots,a_n))=\sum_i {\partial f\over\partial x_i}(x_1,\ldots,x_n) dx_i.$$

   Indeed, taking $f(x_1,x_2)=x_1+x_2$ and $f(x_1,x_2)=x_1 x_2$ recovers the additivity and Leibniz property of derivations, respectively.

   Observe also that $f$ is an element of the free commutative real algebra on $n$ elements, i.e., $\mathbf{R}[x_1,\ldots,x_n]$.

   If we now substitute C^∞-rings for commutative real algebras, we arrive at the correct notion of a derivation for C^∞-rings:

   A __C^∞-derivation__ of a [[C^∞-ring]] $A$ is a map of sets $A\to M$ (where $M$ is a [[module]] over $A$) such that the following chain rule holds for every smooth function $f\in\mathrm{C}^\infty(\mathbf{R}^n)$:
   $$d(f(a_1,\ldots,a_n))=\sum_i {\partial f\over\partial x_i}(x_1,\ldots,x_n) dx_i,$$
   where both sides use the structure of a [[C^∞-ring]] to evaluate a smooth real function on a collection of elements in $A$.
   

   The module of Kähler C^∞-differentials can now be defined in the same manner as ordinary Kähler differentials, using C^∞-derivations instead of ordinary derivations.

   \begin{theorem} (Dubuc, Kock, 1984.) The module of Kähler C^∞-differentials of the C^∞-ring of smooth functions on a smooth manifold $M$ is canonically isomorphic to the module of sections of the cotangent bundle of $M$. \end{theorem}

   Related concepts

   • Kähler differential, derivation

   • C^∞-ring

   References

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

   v1, current