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The de Rham complex (named after Georges de Rham) of a space is the cochain complex that in degree has the differential forms (which may mean: Kähler differential forms) of degree , and whose differential is the de Rham differential or exterior derivative.
As varies this constitutes an abelian sheaf of complexes.
The de Rham complex of a smooth manifold is the cochain complex which in degree has the vector space of degree- differential forms on . The coboundary map is the deRham exterior derivative.
Explicitly, given a differential -form , its de Rham differential can be computed as
where are vector fields on , is the Lie bracket of vector fields, and is the Lie derivative of a smooth function with respect to a vector field .
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 of .
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 , algebraic de Rham cohomology is defined to be the hypercohomology of the de Rham complex .
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 , for 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 .
This is the codomain for the Maurer-Cartan form on in this generality. The shape of is the general Chern character on .
For more on this see at
More precisely, and play the role of the non-negative degree and negative degree part, respectively of the de Rham complex with coefficients in . For more on this see at
This follows from the Mayer-Vietoris sequence associated to the open cover of 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 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 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) and the smooth de Rham complex of :
Here is the simplicial complex corresponding to any smooth triangulation of .
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.