nLab
smooth locus

Context

Synthetic differential geometry

differential geometry

synthetic differential geometry

Axiomatics

Models

Concepts

Theorems

Applications

Contents

Idea

A smooth locus is the formal dual of a finitely generated smooth algebra (or C 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 ( n)/JC^\infty(\mathbb{R}^n)/J, for JJ an ideal of the ordinary underlying algebra.

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

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

Notation

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

Often one also write

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

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

Properties

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

Applications

There are various Grothendieck topologies on 𝕃\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)