nLab
de Rham complex

Context

Cohomology

cohomology

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Differential geometry

synthetic differential geometry

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from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

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Axiomatics

cohesion

  • (shape modality \dashv flat modality \dashv sharp modality)

    (ʃ)(ʃ \dashv \flat \dashv \sharp )

  • dR-shape modality\dashv dR-flat modality

    ʃ dR dRʃ_{dR} \dashv \flat_{dR}

  • tangent cohesion

    • differential cohomology diagram
    • differential cohesion

      • (reduction modality \dashv infinitesimal shape modality \dashv infinitesimal flat modality)

        (&)(\Re \dashv \Im \dashv \&)

      • graded differential cohesion

        • fermionic modality\dashv bosonic modality \dashv rheonomy modality

          (Rh)(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)

        • id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous *

          \array{ && id &\dashv& id \ && \vee && \vee \ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \ && \bot && \bot \ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \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 }

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          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) 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.

          As XX 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 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 (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 TXT X of XX.

          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 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

          For cohesive homotopy types

          In the general context of cohesive homotopy theory in a cohesive (∞,1)-topos H\mathbf{H}, for AHA \in \mathbf{H} a cohesive homotopy type, then the homotopy fiber of the counit of the flat modality

          dRAfib(AA) \flat_{dR} A \coloneqq fib(\flat A \to A)

          may be interpreted as the de Rham complex with coefficients in AA.

          This is the codomain for the Maurer-Cartan form θ ΩA\theta_{\Omega A} on ΩA\Omega A in this generality. The shape of θ ΩA\theta_{\Omega A} is the general Chern character on Π(ΩA)\Pi(\Omega A).

          For more on this see at

          More precisely, dRΣA\flat_{dR} \Sigma A and Π dRΩA\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 Π dRΣA\Pi \flat_{dR} \Sigma A. For more on this see at

          Properties

          Basic theorems

          Relation to Deligne complex

          See at Deligne complex

          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?)

          Last revised on July 26, 2018 at 07:17:53. See the history of this page for a list of all contributions to it.