# nLab de Rham complex

Contents

cohomology

### Theorems

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

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

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# 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 (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).

• 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

## 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

### 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

### 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, $p$-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