group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
Differentials
Tangency
The magic algebraic facts
Theorems
Axiomatics
(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(\esh \dashv \flat \dashv \sharp )$
dR-shape modality$\dashv$ dR-flat modality
$\esh_{dR} \dashv \flat_{dR}$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \Im \dashv \&)$
fermionic modality$\dashv$ bosonic modality $\dashv$ rheonomy modality
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
Models
Models for Smooth Infinitesimal Analysis
smooth algebra ($C^\infty$-ring)
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
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.
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
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 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
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
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
For positive $n$, the de Rham cohomology of the $n$-sphere $S^n$ is
For $n=0$, we have
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.
(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 $X$:
Here $S(X)$ is the simplicial complex corresponding to any smooth triangulation of $X$.
See at Deligne complex
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:
Discussion in algebraic geometry
A useful introduction is
A classical reference on the algebraic version is
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]
Last revised on August 26, 2024 at 07:37:43. See the history of this page for a list of all contributions to it.