nLab
smooth locus

Idea

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

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

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

Notation

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

Often one also write

$$R:=\ell C^{\mathrm{\infty }}(ℝ)$$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

• Models for Smooth Infinitesimal Analysis

References

See the references at C-infinity-ring.