nLab
D-scheme

Context

Geometry

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \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 XX a scheme, analogous to how an XX-scheme is a scheme EXE \to X over XX, a 𝒟 X\mathcal{D}_X-scheme is a scheme over the de Rham space Π inf(X)\mathbf{\Pi}_{inf}(X) of XX.

See also diffiety.

Definition

Definition

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

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

Aff𝒟 XSchemeCMon(𝒟Mod(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\mathcal{D}_X-scheme on a given XX-scheme EXE \to X is the jet bundle of EE.

This is (BeilinsonDrinfeld, 2.3.2).

This fact makes 𝒟\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

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

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