nLab smooth locus

Context

Synthetic differential geometry

differential geometry

synthetic differential geometry

Contents

Idea

A smooth locus is the formal dual of a finitely generated smooth algebra (or ${C}^{\infty }$-ring):

a space that behaves as if its algebra of functions is a finitely generated smooth algebra.

Given that a smooth algebra is a smooth refinement of an ordinary ring with a morphism from $ℝ$, a smooth locus is the analog in well-adapted models for synthetic differential geometry for what in algebraic geometry is an affine variety over $ℝ$.

Definition

A finitely generated smooth algebra is one of the form ${C}^{\infty }\left({ℝ}^{n}\right)/J$, for $J$ an ideal of the ordinary underlying algebra.

Write ${C}^{\infty }{\mathrm{Ring}}^{\mathrm{fin}}$ for the category of finitely generated smooth algebras.

Then the opposite category $𝕃:=\left({C}^{\infty }{\mathrm{Ring}}^{\mathrm{fin}}{\right)}^{\mathrm{op}}$ is the category of smooth loci.

Notation

For $A\in {C}^{\infty }{\mathrm{Ring}}^{\mathrm{fin}}$ one write $\ell A$ for the corresponding object in $𝕃$.

Often one also write

$R:=\ell {C}^{\infty }\left(ℝ\right)$R := \ell C^\infty(\mathbb{R})

for the real line regarded as an object of $𝕃$.

Properties

The category $𝕃$ has the following properties:

• there is a full and faithful functor

$\mathrm{Diff}\to 𝕃$Diff \to \mathbb{L}

from the category Diff of manifolds that preserves pullbacks along transversal maps.

• the Tietze extension theorem holds in $𝕃$: $R$-valued functions on closed subobjects in $𝕃$ have an extension.

Applications

There are various Grothendieck topologies on $𝕃$ and various of its subcategories, such that categories of sheaves on these are smooth toposes that are well-adapted models for synthetic differential geometry.

For more on this see

References

See the references at C-infinity-ring.

Revised on December 15, 2010 11:24:14 by Urs Schreiber (131.211.233.8)