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\begin{document}

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\section*{de Rham complex}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology}

[[!include cohomology - contents]]

\hypertarget{differential_geometry}{}\paragraph*{{Differential geometry}}\label{differential_geometry}

[[!include synthetic differential geometry - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{for_smooth_manifolds}{For smooth manifolds}\dotfill \pageref*{for_smooth_manifolds} \linebreak
\noindent\hyperlink{for_algebraic_objects}{For algebraic objects}\dotfill \pageref*{for_algebraic_objects} \linebreak
\noindent\hyperlink{ForCohesiveHomotopyTypes}{For cohesive homotopy types}\dotfill \pageref*{ForCohesiveHomotopyTypes} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{de_rham_cohomology_of_spheres}{de Rham cohomology of spheres}\dotfill \pageref*{de_rham_cohomology_of_spheres} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{basic_theorems}{Basic theorems}\dotfill \pageref*{basic_theorems} \linebreak
\noindent\hyperlink{relation_to_deligne_complex}{Relation to Deligne complex}\dotfill \pageref*{relation_to_deligne_complex} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{in_differential_geometry}{In differential geometry}\dotfill \pageref*{in_differential_geometry} \linebreak
\noindent\hyperlink{in_algebraic_geometry}{In algebraic geometry}\dotfill \pageref*{in_algebraic_geometry} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The \emph{de Rham complex} (named after [[Georges de Rham]]) $\Omega^\bullet(X)$ of a [[space]] $X$ is the [[cochain complex]] that in degree $n$ has the [[differential form]]s (which may mean: [[Kähler differential form]]s) of degree $n$, and whose [[differential]] is the \textbf{de Rham differential} or \textbf{exterior derivative}.

As $X$ varies this constitutes an [[abelian sheaf]] of complexes.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\hypertarget{for_smooth_manifolds}{}\subsubsection*{{For smooth manifolds}}\label{for_smooth_manifolds}

The \textbf{de Rham complex} of a [[smooth manifold]] is the [[cochain complex]] which in degree $n \in \mathbb{N}$ has the [[vector space]] $\Omega^n(X)$ of degree-$n$ [[differential forms]] on $X$. The coboundary map is the deRham \emph{[[exterior derivative]]}. The [[chain cohomology|cohomology]] of the de Rham complex (hence the [[quotient]] of [[closed differential forms]] by [[exact differential forms]]) is \textbf{de Rham cohomology}. Under the [[wedge product]], the deRham complex becomes a [[differential graded algebra]]. This may be regarded as the [[Chevalley-Eilenberg algebra]] of the [[tangent Lie algebroid]] $T X$ of $X$.

The corresponding [[abelian sheaf]] in this case defines a [[smooth spectrum]] via the [[stable Dold-Kan correspondence]], see at \emph{\href{smooth+spectrum#ExamplesDeRhamSpectra}{smooth spectrum -- Examples -- De Rham spectra}}.

\hypertarget{for_algebraic_objects}{}\subsubsection*{{For algebraic objects}}\label{for_algebraic_objects}

For [[smooth varieties]] $X$, algebraic de Rham cohomology is defined to be the [[hypercohomology]] of the de Rham complex $\Omega_X^\bullet$.

De Rham cohomology has a rather subtle generalization for possibly singular algebraic varieties due to (\hyperlink{Grothendieck}{Grothendieck}).

For [[analytic spaces]]

\begin{itemize}%
\item T. Bloom, M. Herrera, \emph{De Rham cohomology of an analytic space}, Inv. Math. \textbf{7} (1969), 275-296, \href{http://dx.doi.org/10.1007/BF01425536}{doi}

\end{itemize}
\hypertarget{ForCohesiveHomotopyTypes}{}\subsubsection*{{For cohesive homotopy types}}\label{ForCohesiveHomotopyTypes}

In the general context of [[cohesive homotopy theory]] in a [[cohesive (∞,1)-topos]] $\mathbf{H}$, for $A \in \mathbf{H}$ a [[cohesive homotopy type]], then the [[homotopy fiber]] of the [[counit of a comonad|counit]] of the [[flat modality]]

\begin{displaymath}
\flat_{dR} A \coloneqq fib(\flat A \to A)
\end{displaymath}
may be interpreted as the de Rham complex with [[coefficients]] in $A$.

This is the [[codomain]] for the [[Maurer-Cartan form]] $\theta_{\Omega A}$ on $\Omega A$ in this generality. The [[shape modality|shape]] of $\theta_{\Omega A}$ is the general [[Chern character]] on $\Pi(\Omega A)$.

For more on this see at

\begin{itemize}%
\item \emph{\href{cohesive+%28infinity%2C1%29-topos+--+structures#deRhamCohomology}{structures in a cohesive infinity-topos -- de Rham cohomology}}

\end{itemize}
More precisely, $\flat_{dR} \Sigma A$ and $\Pi_{dR} \Omega A$ play the role of the non-negative degree and negative degree part, respectively of the de Rham complex with coefficients in $\Pi \flat_{dR} \Sigma A$. For more on this see at

\begin{itemize}%
\item \emph{\href{differential%20cohomology%20diagram#DeRhamCoefficients}{differential cohomology diagram -- de Rham coefficients}}.

\end{itemize}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{de_rham_cohomology_of_spheres}{}\subsubsection*{{de Rham cohomology of spheres}}\label{de_rham_cohomology_of_spheres}

\begin{prop}
\label{}\hypertarget{}{}
For positive $n$, the de Rham cohomology of the $n$-[[sphere]] $S^n$ is

\begin{displaymath}
H^p(S^n)
  = 
  \left\{
    \itexarray{
      \mathbb{R} & if\; p = 0,n  
      \\
      0 & otherwise
    }
  \right.
  \,.
\end{displaymath}
For $n=0$, we have

\begin{displaymath}
H^p(S^0)
  = 
  \left\{
    \itexarray{
      \mathbb{R} \oplus \mathbb{R} & if\; p = 0
      \\
      0 & otherwise
    }
  \right.
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
This follows from the [[Mayer-Vietoris sequence]] associated to the open cover of $S^n$ by the subset excluding just the north pole and the subset excluding just the south pole, together with the fact that the dimension of the $0^{th}$ de Rham cohomology of a smooth manifold is its number of connected components.

\end{proof}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{basic_theorems}{}\subsubsection*{{Basic theorems}}\label{basic_theorems}

\begin{itemize}%
\item [[Poincare lemma]]


\item [[de Rham theorem]]



\end{itemize}
\hypertarget{relation_to_deligne_complex}{}\subsubsection*{{Relation to Deligne complex}}\label{relation_to_deligne_complex}

See at \emph{[[Deligne complex]]}

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[pullback of differential forms]]


\item [[equivariant de Rham cohomology]]


\item [[de Rham-Witt complex]]


\item [[twisted de Rham cohomology]]


\item [[Deligne cohomology]]


\item [[de Rham space]]


\item [[Lie algebra valued differential forms]]


\item [[L-infinity algebra valued differential forms]]


\item [[absolute de Rham cohomology]]


\item [[Dolbeault complex]], [[Dolbeault cohomology]]


\item [[holomorphic de Rham complex]]


\item [[Hodge-de Rham spectral sequence]]


\item [[chiral de Rham complex]]


\item [[crystalline cohomology]], [[comparison theorem (crystalline cohomology)]]


\item [[Hodge cohomology]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\hypertarget{in_differential_geometry}{}\subsubsection*{{In differential geometry}}\label{in_differential_geometry}

(\ldots{})

\hypertarget{in_algebraic_geometry}{}\subsubsection*{{In algebraic geometry}}\label{in_algebraic_geometry}

A useful introduction is

\begin{itemize}%
\item Kiran Kedlaya, \emph{$p$-adic cohomology, from theory to practice} (\href{http://swc.math.arizona.edu/aws/2007/KedlayaNotes11Mar.pdf}{pdf})

\end{itemize}
A classical reference on the algebraic version is

\begin{itemize}%
\item [[Alexander Grothendieck]], \emph{On the De Rham cohomology of algebraic varieties}, Publications Math\'e{}matiques de l'IH\'E{}S \textbf{29}, 351-359 (1966), \href{http://www.numdam.org/item?id=PMIHES_1966__29__95_0}{numdam}.

\end{itemize}
\begin{itemize}%
\item A. Grothendieck, \emph{Crystals and the de Rham cohomology of schemes}, in Giraud, Jean; Grothendieck, Alexander; Kleiman, Steven L. et al., Dix Expos\'e{}s sur la Cohomologie des Sch\'e{}mas, Advanced studies in pure mathematics \textbf{3}, Amsterdam: North-Holland, pp. 306--358, \href{http://www.ams.org/mathscinet-getitem?mr=0269663}{MR0269663}, \href{http://www.math.jussieu.fr/~leila/grothendieckcircle/DixExp.pdf}{pdf}
\item [[Robin Hartshorne]], \emph{On the de Rham cohomology of algebraic varieties}, Publ. Math\'e{}matiques de l'IH\'E{}S \textbf{45} (1975), p. 5-99 \href{http://www.ams.org/mathscinet-getitem?mr=55:5633}{MR55\#5633}
\item P. Monsky, \emph{Finiteness of de Rham cohomology}, Amer. J. Math. \textbf{94} (1972), 237--245, \href{http://www.ams.org/mathscinet-getitem?mr=301017}{MR301017}, \href{http://dx.doi.org/10.2307/2373603}{doi}

\end{itemize}
See also

\begin{itemize}%
\item Yves Andr\'e{}, \emph{Comparison theorems between algebraic and analytic De Rham cohomology} (\href{http://www.emis.de/journals/JTNB/2004-2/pages335-355.pdf}{pdf})


\item [[Mikhail Kapranov]], \emph{DG-Modules and vanishing cycles} ([[KapranovDGModuleVanishingCycle.pdf:file]])



\end{itemize}
[[!redirects deRham complex]] [[!redirects deRham algebra]] [[!redirects de Rham algebra]] [[!redirects deRham dga]] [[!redirects de Rham dga]] [[!redirects deRham dg-algebra]] [[!redirects de Rham dg-algebra]] [[!redirects de Rham cohomology]] [[!redirects deRham cohomology]]

[[!redirects de Rham complex]]

[[!redirects de Rham complexes]]



\end{document}