nLab
de Rham space

Context

Synthetic differential geometry

differential geometry

synthetic differential geometry

Axiomatics

Models

Concepts

Theorems

Applications

Discrete and concrete objects

Contents

Idea

In a context of synthetic differential geometry or D-geometry, the de Rham space dR(X)dR(X) of a space XX is the quotient of XX that identifies infinitesimally close points.

It is the coreduced reflection of XX.

Definition

On Rings opRings^{op}

Let CRing be the category of commutative rings. For RCRingR \in CRing, write IRI \in R for the nilradical of RR, the ideal consisting of the nilpotent elements. The canonical projection RR/IR \to R/I to the quotient y the ideal corresponds in the opposite category Ring opRing^{op} to the inclusion

Spec(R/I)SpecR Spec (R/I) \to Spec R

of the reduced part of SpecRSpec R.

Definition

For XPSh(Ring op)X \in PSh(Ring^{op}) a presheaf on Ring opRing^{op} (for instance a scheme), its de Rham space X dRX_{dR} is the presheaf defined by

X dR:SpecRX(Spec(R/I)). X_{dR} : Spec R \mapsto X\left(Spec \left(R/I\right)\right) \,.

Properties

As a quotient

Proposition

If XPSh(Ring op)X \in PSh(Ring^{op}) is a smooth scheme then the canonical morphism

XX dR X \to X_{dR}

is an epimorphism (hence an epimorphism over each SpecRSpec R) and therefore in this case X dRX_{dR} is the quotient of the relation “being infinitesimally close” between points of XX: we have that X dRX_{dR} is the coequalizer

X dR=lim (X infX), X_{dR} = \lim_\to \left( X^{inf} \stackrel{\to}{\to} X \right) \,,

of the two projections out of the formal neighbourhood of the diagonal.

Relation to jet bundles

For EXE \to X a bundle over XX, its direct image under base change along the projection map XΠ infXX \longrightarrow \Pi_{inf} X yields its jet bundle. See there for more.

Remark

In terms of differential homotopy type theory this means that forming “jet types” of dependent types over XX is the dependent product operation along the unit of the infinitesimal shape modality

jet(E)XΠ infXE. jet(E) \coloneqq \underset{X \to \Pi_{inf}X}{\prod} E \,.

Relation to formally étale morphism of schemes

Crystalline site

For X:RingSetX : Ring \to Set a scheme, the big site Ring op/X dRRing^{op}/X_{dR} of X dRX_{dR}, is the crystaline site of XX.

Grothendieck connection

Morphisms X dRModX_{dR} \to Mod encode flat higher connections: local systems.

Accordingly, descent for deRham spaces – sometimes called deRham descent encodes flat 1-connections. This is described at Grothendieck connection,

D-modules

The category of D-modules on a space is equivalent to that of quasicoherent sheaves on the corresponding deRham space.

Accordingly, quasicoherent \infty-stacks on the full Π inf(X)\Pi^{inf}(X) encode a higher categorical version of this, as discussed at ∞-vector bundle.

Infinitesimal path \infty-groupoids

References

The term de Rham space or de Rham stack apparently goes back to

  • Carlos Simpson, Homotopy over the complex numbers and generalized de Rham cohomology Moduli of VectorBundles, M. Maruyama, ed., Dekker (1996), 229-263.

A review of the constructions is on the first two pages of

  • Jacob Lurie, Notes on crystals and algebraic 𝒟\mathcal{D}-modules (pdf)

The deRham space construction on spaces (schemes) is described in section 3, p. 7

which goes on to assert the existence of its derived functor on the homotopy category HoSh (C)Ho Sh_\infty(C) of ∞-stacks in proposition 3.3. on the same page.

The characterization of formally smooth scheme as above is also on that page.

See also online comments by David Ben-Zvi here and here on the nnCafé. and here on MO.

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Revised on April 25, 2014 03:58:41 by Urs Schreiber (88.128.80.141)