nLab D-scheme

Contents

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

tangent cohesion

differential cohesion

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

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.

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)

Last revised on May 15, 2015 at 17:40:06. See the history of this page for a list of all contributions to it.