nLab de Rham space

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 by 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

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

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

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

Relation to formally étale morphisms of schemes

• formally étale morphism of schemes

Related concepts

Crystalline site

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

Grothendieck connection

Morphisms $X_{dR} \to Mod$ encode flat higher connections: local systems.

Accordingly, descent for de Rham spaces – sometimes called de Rham 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 de Rham space (Lurie, above theorem 0.4).

Infinitesimal path $\infty$-groupoids

• infinitesimal path ∞-groupoid

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, Toulouse preprint no. 50, April 1995, In: M. Maruyama, (ed.) Moduli of Vector Bundles (Taniguchi symposium December 1994), Lecture notes in Pure and Applied Mathematics, Issue 189, Page 229-264, Dekker (1996), 229-263 (web)

But actually there it just has the notation “$X_{dR}$” and then the functor it co-represents is called the “de Rham shape” of $X$.

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

• Jacob Lurie, Notes on crystals and algebraic $\mathcal{D}$-modules (2010) [pdf]

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

• Carlos Simpson, Constantin Teleman, de Rham theorem for $\infty$-stacks, 1997 (pdf, pdf)

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.

Similar discussion in a context of derived algebraic geometry is in

• Dennis Gaitsgory, Nick Rozenblyum, section 1 of Crystals and D-modules (pdf)

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.