# 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 $\mathrm{dR}\left(X\right)$ of a space $X$ is the quotient of $X$ that identifies infinitesimally close points.

It is the coreduced reflection of $X$.

## Definition

### On ${\mathrm{Rings}}^{\mathrm{op}}$

Let CRing be the category of commutative rings. For $R\in \mathrm{CRing}$, write $I\in R$ for the nilradical of $R$, the ideal consisting of the nilpotent elements. The canonical projection $R\to R/I$ corresponds in the opposite category ${\mathrm{Ring}}^{\mathrm{op}}$ to the inclusion

$\mathrm{Spec}R/I\to \mathrm{Spec}R\phantom{\rule{thinmathspace}{0ex}}.$Spec R/I \to Spec R \,.
###### Definition

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

${X}_{\mathrm{dR}}:\mathrm{Spec}R↦X\left(\mathrm{Spec}R/I\right)\phantom{\rule{thinmathspace}{0ex}}.$X_{dR} : Spec R \mapsto X(Spec R/I) \,.

## Properties

###### Proposition

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

$X\to {X}_{\mathrm{dR}}$X \to X_{dR}

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

${X}_{\mathrm{dR}}=\underset{\to }{\mathrm{lim}}\left({X}^{\mathrm{inf}}\stackrel{\to }{\to }X\right)\phantom{\rule{thinmathspace}{0ex}},$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 $X:\mathrm{Ring}\to \mathrm{Set}$ a scheme, the big site ${\mathrm{Ring}}^{\mathrm{op}}/{X}_{\mathrm{dR}}$ of ${X}_{\mathrm{dR}}$, is the crystaline site of $X$.

### Grothendieck connection

Morphisms ${X}_{\mathrm{dR}}\to \mathrm{Mod}$ 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 ${\Pi }^{\mathrm{inf}}\left(X\right)$ 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 $𝒟$-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 $\mathrm{Ho}{\mathrm{Sh}}_{\infty }\left(C\right)$ 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 5, 2013 21:54:00 by Urs Schreiber (89.204.138.93)