# 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 $\mathbb{R}$, a smooth locus is the analog in well-adapted models for synthetic differential geometry for what in algebraic geometry is an affine variety over $\mathbb{R}$.

## Definition

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

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

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

### Notation

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

Often one also write

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

for the real line regarded as an object of $\mathbb{L}$.

## Properties

The category $\mathbb{L}$ has the following properties:

• there is a full and faithful functor

$Diff \to \mathbb{L}$

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

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

## Applications

There are various Grothendieck topologies on $\mathbb{L}$ 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)