Definition

For smooth manifolds

The de Rham complex of a smooth manifold is the cochain complex which in degree n∈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) iℒ v iω(v 0,…,v i−1,v i+1,…,v k)+∑ i<j(−1) i+jω([v i,v j],v 0,…,v i−1,v i+1,…,v j−1,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 ℒ 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 A∈HA \in \mathbf{H} a cohesive homotopy type, then the homotopy fiber of the counit of the flat modality

♭ dRA≔fib(♭A→A) \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

structures in a cohesive infinity-topos – de Rham cohomology

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

differential cohomology diagram – de Rham coefficients.

Examples

de Rham cohomology of spheres

Proposition

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

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

Proof

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.

Properties

Basic theorems

Relation to PL de Rham complex

Relation to Deligne complex

See at Deligne complex

References

In differential geometry

Discussion in differential geometry:

Raoul Bott, Loring Tu, Differential Forms in Algebraic Topology, Graduate Texts in Mathematics 82, Springer 1982 (doi:10.1007/978-1-4757-3951-0)
Georges de Rham, Chapter II of: Differentiable Manifolds – Forms, Currents, Harmonic Forms, Grundlehren 266, Springer (1984) [doi:10.1007/978-3-642-61752-2]
Dominic G. B. Edelen, Applied exterior calculus, Wiley (1985) [GoogleBooks]

With an eye towards application in mathematical physics:

Mikio Nakahara, Chapter 6 of: Geometry, Topology and Physics, IOP 2003 (doi:10.1201/9781315275826, pdf)

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