nLab
de Rham complex

Idea

The de Rham complex ${\Omega }^{•}(X)$ 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.

(named after Georges de Rham)

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. The cohomology of the de Rham complex 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 $TX$ of $X$.

For algebraic objects

For smooth varieties $X$, algebraic de Rham cohomology is defined to be the hypercohomology of the de Rham complex ${\Omega }_X^{•}$.

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

Properties

• Poincare lemma

• de Rham theorem

Related concepts

• pullback of differential forms

• 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

• Hodge-de Rham spectral sequence?

References

In differential geometry

(…)

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 and vanishing cycles (KapranovDGModuleVanishingCycle.pdf?)