Given that a smooth algebra is a smooth refinement of an ordinary ring with a morphism from , a smooth locus is the analog in well-adapted models for synthetic differential geometry for what in algebraic geometry is an affine variety over .
A finitely generated smooth algebra is one of the form , for an ideal of the ordinary underlying algebra.
Write for the category of finitely generated smooth algebras.
For one write for the corresponding object in .
Often one also write
for the real line regarded as an object of .
The category has the following properties:
there is a full and faithful functor
the Tietze extension theorem holds in : -valued functions on closed subobjects in have an extension.
There are various Grothendieck topologies on 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
See the references at C-infinity-ring.