nLab
de Rham space

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Synthetic differential geometry

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synthetic differential geometry

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coreduced object

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Idea

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

It is the coreduced reflection of X.

Definition

On Rings op

Let CRing be the category of commutative rings. For R∈CRing, write I∈R for the nilradical of R, the ideal consisting of the nilpotent elements. The canonical projection R→R/I corresponds in the opposite category Ring op to the inclusion

SpecR/I→SpecR.Spec R/I \to Spec R \,.
Definition

For X∈PSh(Ring op) a presheaf on Ring op (for instance a scheme), its de Rham space X dR is the presheaf defined by

X dR:SpecR↦X(SpecR/I).X_{dR} : Spec R \mapsto X(Spec R/I) \,.

Properties

Proposition

If X∈PSh(Ring op) is a smooth scheme then the canonical morphism

X→X dRX \to X_{dR}

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

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

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

Crystalline site

For X:Ring→Set a scheme, the big site Ring op/X dR of X dR, is the crystaline site of X.

Grothendieck connection

Morphisms X dR→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 ∞-stacks on the full Π inf(X) encode a higher categorical version of this, as discussed at ∞-vector bundle.

Infinitesimal path ∞-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

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) 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 nCafé. and here on MO.

Revised on January 5, 2013 21:54:00 by Urs Schreiber (89.204.138.93)
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