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See the article Kähler C^∞-differentials of smooth functions are differential 1-forms for the necessary background for this article, including the notions of C^∞-ring, C^∞-derivation, and Kähler C^∞-differential.
In algebraic geometry, (algebraic) differential forms on the Zariski spectrum of a [commutative ring R (or a commutative k-algebra R) can be defined as the free commutative differential graded algebra on R.
This definition does not quite work for smooth manifolds: as already explained in the article Kähler C^∞-differentials of smooth functions are differential 1-forms, the notion of a Kähler differential must be refined in order to extract smooth differential 1-forms from the C^∞-ring of smooth functions on a smooth manifold M.
Thus, in order to get the algebra of smooth differential forms, the notion of a commutative differential graded algebra must likewise be adjusted.
\begin{definition} A commutative differential graded C^∞-ring is a real commutative differential graded algebra A whose degree 0 component A0 is equipped with a structure of a C^∞-ring in such a way that the degree 0 differential A0→A1 is a C^∞-derivation. \end{definition}
With this definition, we can recover smooth differential forms in a manner similar to algebraic geometry, deducing the following consequence of the Dubuc–Kock theorem for Kähler C^∞-differentials.
\begin{theorem} The free commutative differential graded C^∞-ring on the C^∞-ring of smooth functions on a smooth manifold M is canonically isomorphic to the differential graded algebra of smooth differential forms on M. \end{theorem}
The Poincaré lemma becomes a trivial consequence of the above theorem.
\begin{proposition} For every n≥0, the canonical map
R[0]→Ω(Rn)is a quasi-isomorphism of differential graded algebras. \end{proposition}
\begin{proof} (Copied from the MathOverflow answer.) The de Rham complex of a finite-dimensional smooth manifold M is the free C^∞-dg-ring on the C^∞-ring C∞(M). If M is the underlying smooth manifold of a finite-dimensional real vector space V, then C∞(M) is the free C^∞-ring on the vector space V* (the real dual of V). Thus, the de Rham complex of a finite-dimensional real vector space V is the free C^∞-dg-ring on the vector space V*. This free C^∞-dg-ring is the free C^∞-dg-ring on the free cochain complex on the vector space V*. The latter cochain complex is simply V*→V* with the identity differential. It is cochain homotopy equivalent to the zero cochain complex, and the free functor from cochain complexes to C^∞-dg-rings preserves cochain homotopy equivalences. Thus, the de Rham complex of the smooth manifold V is cochain homotopy equivalent to the free C^∞-dg-ring on the zero cochain complex, i.e., R in degree 0. \end{proof}
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