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D-scheme

Idea

For $X$ a scheme, analogous to how an $X$-scheme is a scheme $E\to X$ over $X$, a ${𝒟}_X$-scheme is a scheme over the de Rham space ${\Pi }_{\inf }(X)$ of $X$.

See also diffiety.

Definition

Definition

For $X$ a scheme, a ${𝒟}_X$-scheme is a scheme $E\to {\Pi }_{\inf }(X)$ over the de Rham space ${\Pi }_{\inf }(X)$ of $X$.

Remark

This definition makes sense in much greater generality: in any context of infinitesimal cohesion.

Properties

Relation to D-modules

Definition

In the sheaf topos over affine schemes, an $X$-affine ${𝒟}_X$-scheme is a commutative monoid object in the monoidal category of quasicoherent sheaves $\mathrm{QC}({\Pi }_{\inf }(X))$, which is equivalently the category of D-modules over $X$:

$$\mathrm{Aff}{𝒟}_X\mathrm{Scheme}\simeq \mathrm{CMon}(𝒟\mathrm{Mod}(X)).$$Aff \mathcal{D}_X Scheme \simeq CMon(\mathcal{D}Mod(X)) \,.

This is (BeilinsonDrinfeld, section 2.3).

Proposition

This is indeed equivalent to the above abstract definition

This appears as (Lurie, theorem, 0.6 and below remark 0.7)

Relation to jet schemes

The free ${𝒟}_X$-scheme on a given $X$-scheme $E\to X$ is the jet bundle of $E$.

This is (BeilinsonDrinfeld, 2.3.2).

This fact makes $𝒟$-geometry a natural home for variational calculus.

Related concepts

• scheme

• D-module

• de Rham space

• diffiety

References

The definition in terms of monoids in D-modules is in section 2.3 in

• Alexander Beilinson and Vladimir Drinfeld, Chiral Algebras

The observation that this is equivalent to the abstract definition given above appears in pages 5 and 6 of

• Jacob Lurie, Notes on crystals and algebraic $𝒟$-modules