nLab
de Rham complex

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Differential geometry

differential geometry

synthetic differential geometry

Axiomatics

Models

Concepts

Theorems

Applications

Contents

Idea

The de Rham complex Ω (X)\Omega^\bullet(X) of a space XX is the cochain complex that in degree nn has the differential forms (which may mean: Kähler differential forms) of degree nn, 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 nn \in \mathbb{N} has the vector space Ω n(X)\Omega^n(X) of degree-nn differential forms on XX. 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 TXT X of XX.

For algebraic objects

For smooth varieties XX, algebraic de Rham cohomology is defined to be the hypercohomology of the de Rham complex Ω X \Omega_X^\bullet.

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

References

In differential geometry

(…)

In algebraic geometry

A useful introduction is

  • Kiran Kedlaya, pp-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?)

Revised on November 21, 2013 22:40:06 by Urs Schreiber (82.169.114.243)