# nLab de Rham complex

cohomology

### Theorems

#### Differential geometry

differential geometry

synthetic differential geometry

# Contents

## 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. 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 $T X$ 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).

• 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

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

## Properties

### Relation to Deligne complex

See at Deligne complex

## References

(…)

### 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