nLab de Rham complex





Special and general types

Special notions


Extra structure



Differential geometry

synthetic differential geometry


from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry



smooth space


The magic algebraic facts




infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }


Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)



The de Rham complex (named after Georges de Rham) Ω (X)\Omega^\bullet(X) of a space XX is the cochain complex that in degree nn has the differential forms (which may mean: Kähler differential forms) of degree nn, and whose differential is the de Rham differential or exterior derivative.

As XX varies this constitutes an abelian sheaf of complexes.


For smooth manifolds

The de Rham complex of a smooth manifold is the cochain complex which in degree nn \in \mathbb{N} has the vector space Ω n(X)\Omega^n(X) of degree-nn differential forms on XX. The coboundary map is the deRham exterior derivative.

Explicitly, given a differential kk-form ω\omega, its de Rham differential dωd\omega can be computed as

dω(v 0,,v k)= i(1) i v iω(v 0,,v i1,v i+1,,v k)+ i<j(1) i+jω([v i,v j],v 0,,v i1,v i+1,,v j1,v j+1,,v k),d\omega(v_0,\ldots,v_k)=\sum_i (-1)^i \mathcal{L}_{v_i} \omega(v_0,\ldots,v_{i-1},v_{i+1},\ldots,v_k)+\sum_{i\lt j}(-1)^{i+j}\omega([v_i,v_j],v_0,\ldots,v_{i-1},v_{i+1},\ldots,v_{j-1},v_{j+1},\ldots,v_k),

where v iv_i are vector fields on XX, [,][-,-] is the Lie bracket of vector fields, and v\mathcal{L}_{v} is the Lie derivative of a smooth function with respect to a vector field vv.

The cohomology of the de Rham complex (hence the quotient of closed differential forms by exact differential forms) is 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 TXT X of XX.

The corresponding abelian sheaf in this case defines a smooth spectrum via the stable Dold-Kan correspondence, see at smooth spectrum – Examples – De Rham spectra.

For algebraic objects

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

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

For analytic spaces

  • T. Bloom, M. Herrera, De Rham cohomology of an analytic space, Inv. Math. 7 (1969), 275-296, doi

For cohesive homotopy types

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

dRAfib(AA) \flat_{dR} A \coloneqq fib(\flat A \to A)

may be interpreted as the de Rham complex with coefficients in AA.

This is the codomain for the Maurer-Cartan form θ ΩA\theta_{\Omega A} on ΩA\Omega A in this generality. The shape of θ ΩA\theta_{\Omega A} is the general Chern character on Π(ΩA)\Pi(\Omega A).

For more on this see at

More precisely, dRΣA\flat_{dR} \Sigma A and Π dRΩA\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 Π dRΣA\Pi \flat_{dR} \Sigma A. For more on this see at


de Rham cohomology of spheres


For positive nn, the de Rham cohomology of the nn-sphere S nS^n is

H p(S n)={ ifp=0,n 0 otherwise. H^p(S^n) = \left\{ \array{ \mathbb{R} & if\; p = 0,n \\ 0 & otherwise } \right. \,.

For n=0n=0, we have

H p(S 0)={ ifp=0 0 otherwise. H^p(S^0) = \left\{ \array{ \mathbb{R} \oplus \mathbb{R} & if\; p = 0 \\ 0 & otherwise } \right. \,.

This follows from the Mayer-Vietoris sequence associated to the open cover of S nS^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 th0^{th} de Rham cohomology of a smooth manifold is its number of connected components.


Basic theorems

Relation to PL de Rham complex


(PL de Rham complex of smooth manifold is equivalent to de Rham complex)

Let XX be a smooth manifold.

We have the following zig-zag of dgc-algebra quasi-isomorphisms between the PL de Rham complex of (the topological space underlying) XX and the smooth de Rham complex of XX:

Ω PLdR (S(X)) Ω dR (X) i * i poly p * Ω PLdR (X)=Ω PLdR (Sing(X)) Ω PSdR (S(X)) \array{ && \Omega^\bullet_{PLdR} \big( S(X) \big) && && \Omega^\bullet_{dR}(X) \\ & {}^{ \mathllap{ i^\ast } } \nearrow & & \searrow^{ \mathrlap{ i_{poly} } } & & {}^{ \mathllap{ p^\ast } } \swarrow \\ \mathllap{ \Omega^\bullet_{PLdR}(X) \;=\; } \Omega^\bullet_{PLdR} \big( Sing(X) \big) && && \Omega^\bullet_{PSdR} \big( S(X) \big) }

Here S(X)S(X) is the simplicial complex corresponding to any smooth triangulation of XX.

Relation to Deligne complex

See at Deligne complex


In differential geometry

Discussion in differential geometry:

With an eye towards application in mathematical physics:

In algebraic geometry

Discussion in algebraic geometry

A useful introduction is

  • Kiran Kedlaya, pp-adic cohomology, from theory to practice (pdf)

A classical reference on the algebraic version is

  • Alexander Grothendieck, On the De Rham cohomology of algebraic varieties, Publications Mathématiques de l’IHÉS 29, 351-359 (1966), numdam.
  • A. Grothendieck, Crystals and the de Rham cohomology of schemes, in Giraud, Jean; Grothendieck, Alexander; Kleiman, Steven L. et al., Dix Exposés sur la Cohomologie des Schémas, Advanced studies in pure mathematics 3, Amsterdam: North-Holland, pp. 306–358, MR0269663, pdf
  • Robin Hartshorne, On the de Rham cohomology of algebraic varieties, Publ. Mathématiques de l’IHÉS 45 (1975), p. 5-99 MR55#5633
  • P. Monsky, Finiteness of de Rham cohomology, Amer. J. Math. 94 (1972), 237–245, MR301017, doi

See also

  • Yves André, Comparison theorems between algebraic and analytic De Rham cohomology (pdf)

  • Mikhail Kapranov, DG-Modules over the de Rham complex and the vanishing cycles functor, Lecture Notes in Mathematics 1479, Springer (1991) [doi:10.1007/BFb0086264]

Last revised on January 28, 2024 at 17:05:25. See the history of this page for a list of all contributions to it.