Synthetic differential geometry
Discrete and concrete objects
In a context of synthetic differential geometry or D-geometry, the de Rham space of a space is the quotient of that identifies infinitesimally close points.
It is the coreduced reflection of .
Let CRing be the category of commutative rings. For , write for the nilradical of , the ideal consisting of the nilpotent elements. The canonical projection to the quotient y the ideal corresponds in the opposite category to the inclusion
of the reduced part of .
For a presheaf on (for instance a scheme), its de Rham space is the presheaf defined by
As a quotient
If is a smooth scheme then the canonical morphism
is an epimorphism (hence an epimorphism over each ) and therefore in this case is the quotient of the relation “being infinitesimally close” between points of : we have that is the coequalizer
of the two projections out of the formal neighbourhood of the diagonal.
Relation to jet bundles
For a bundle over , its direct image under base change along the projection map yields its jet bundle. See there for more.
Relation to formally étale morphism of schemes
For a scheme, the big site of , is the crystaline site of .
Morphisms 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,
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 encode a higher categorical version of this, as discussed at ∞-vector bundle.
Infinitesimal path -groupoids
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 -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 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 Café. and here on MO.