Standard smooth algebras #
A standard smooth algebra is an algebra that admits a SubmersivePresentation
. A standard
smooth algebra is of relative dimension n
if it admits a submersive presentation of
dimension n
.
While every standard smooth algebra is smooth, the converse does not hold. But if S
is R
-smooth,
then S
is R
-standard smooth locally on S
, i.e. there exists a set { t }
of S
that
generates the unit ideal, such that Sₜ
is R
-standard smooth for every t
(TODO, see below).
Main definitions #
All of these are in the Algebra
namespace. Let S
be an R
-algebra.
Algebra.IsStandardSmooth
:S
isR
-standard smooth ifS
admits a submersiveR
-presentation.Algebra.IsStandardSmooth.relativeDimension
: IfS
isR
-standard smooth this is the dimension of an arbitrary submersiveR
-presentation ofS
. This is independent of the choice of the presentation (TODO, see below).Algebra.IsStandardSmoothOfRelativeDimension n
:S
isR
-standard smooth of relative dimensionn
if it admits a submersiveR
-presentation of dimensionn
.
TODO #
- Show that locally on the target, smooth algebras are standard smooth.
Notes #
This contribution was created as part of the AIM workshop "Formalizing algebraic geometry" in June 2024.
The relative dimension of a standard smooth R
-algebra S
is
the dimension of an arbitrarily chosen submersive R
-presentation of S
.
Note: If S
is non-trivial, this number is independent of the choice of the presentation as it is
equal to the S
-rank of Ω[S/R]
(see IsStandardSmoothOfRelativeDimension.rank_kaehlerDifferential
).