nLab de Rham complex

Idea

The 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 forms (which may mean: Kähler differential forms) of degree $n$, and whose differential is the de Rham differential or exterior derivative.

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

Definition

For smooth manifolds

The 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 exterior derivative.

Explicitly, given a differential $k$-form $\omega$, its de Rham differential $d\omega$ can be computed as

$$d\omega(v_0,\ldots,v_k)=\sum_i (-1)^i \mathcal{L}_{v_i} \omega(v_0,\ldots,v_{i-1},v_{i+1},\ldots,v_k)+\sum_{i\lt j}(-1)^{i+j}\omega([v_i,v_j],v_0,\ldots,v_{i-1},v_{i+1},\ldots,v_{j-1},v_{j+1},\ldots,v_k),$$

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

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 $T X$ of $X$.

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 $X$, 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 (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 $\mathbf{H}$, for $A \in \mathbf{H}$ a cohesive homotopy type, then the homotopy fiber of the counit of the flat modality

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

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 of $\theta_{\Omega A}$ is the general Chern character on $\Pi(\Omega A)$.

For more on this see at

• structures in a cohesive infinity-topos – de Rham cohomology

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

• differential cohomology diagram – de Rham coefficients.

Examples

de Rham cohomology of spheres

Proposition

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

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

For $n=0$, we have

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

Proof

This follows from the Mayer-Vietoris sequence associated to the open cover of $S^n$ 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.

Properties

Basic theorems

• Poincare lemma

• de Rham theorem

• nonabelian de Rham cohomology

Relation to PL de Rham complex

Proposition

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

Let $X$ 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) $X$ and the smooth de Rham complex of $X$:

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

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

Relation to Deligne complex

See at Deligne complex

Related concepts

• pullback of differential forms

• equivariant de Rham cohomology

• de Rham-Witt complex

• twisted de Rham cohomology

• Deligne cohomology

• de Rham space

• Lie algebra valued differential forms

• L-infinity algebra valued differential forms

• absolute de Rham cohomology

• Dolbeault complex, Dolbeault cohomology

• holomorphic de Rham complex

• Hodge-de Rham spectral sequence

• chiral de Rham complex

• crystalline cohomology, comparison theorem (crystalline cohomology)

• Hodge cohomology

• PL de Rham 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, $p$-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]