# nLab de Rham space

### Context

#### Synthetic differential geometry

differential geometry

synthetic differential geometry

# Contents

## 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 $R \in CRing$, write $I \in R$ for the nilradical of $R$, the ideal consisting of the nilpotent elements. The canonical projection $R \to R/I$ to the quotient y the ideal corresponds in the opposite category $Ring^{op}$ to the inclusion

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

of the reduced part of $Spec R$.

###### Definition

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

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

## Properties

### As a quotient

###### Proposition

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

$X \to X_{dR}$

is an epimorphism (hence an epimorphism over each $Spec R$) 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

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

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

### Relation to jet bundles

For $E \to X$ a bundle over $X$, its direct image under base change along the projection map $X \longrightarrow \Pi_{inf} X$ 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 $X$ is the dependent product operation along the unit of the infinitesimal shape modality

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

### Crystalline site

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

### Grothendieck connection

Morphisms $X_{dR} \to Mod$ 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 $\Pi^{inf}(X)$ encode a higher categorical version of this, as discussed at ∞-vector bundle.

## 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 $\mathcal{D}$-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 $Ho Sh_\infty(C)$ 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 $n$Café. and here on MO.

Revised on January 28, 2014 08:54:53 by Urs Schreiber (82.113.121.62)