# nLab D-scheme

Contents

### Context

Ingredients

Concepts

Constructions

Examples

Theorems

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

tangent cohesion

differential cohesion

singular cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# 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.

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)