# nLab D-scheme

## Theorems

#### Differential geometry

differential geometry

synthetic differential geometry

# Contents

## Idea

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

## Definition

###### Definition

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

###### Remark

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

## Properties

### Relation to D-modules

###### Definition

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

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

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

This is (BeilinsonDrinfeld, 2.3.2).

This fact makes $\mathcal{D}$-geometry a natural home for variational calculus.

## References

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

• chapter 2, Geometry of D-schemes (pdf)

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 $\mathcal{D}$-modules, 2009 (pdf)

Revised on May 15, 2015 17:40:06 by Urs Schreiber (195.113.30.252)