Smoothness after inverting a Jacobian minor #
A finite presentation equipped with a choice of as many variables as relations becomes standard smooth after inverting the corresponding Jacobian determinant. Its relative dimension is the number of variables minus the number of relations. This provides smooth coordinate charts for hypersurfaces.
For a hypersurface, one can apply the criterion wherever a chosen partial derivative is invertible; with several relations, use the determinant of a square Jacobian minor.
References #
- Stacks Project, Tag 00T7.
theorem
Algebra.PreSubmersivePresentation.isStandardSmoothOfRelativeDimension_localizationAway
{R : Type u_1}
{S : Type u_2}
{ι : Type u_3}
{σ : Type u_4}
[CommRing R]
[CommRing S]
[Algebra R S]
[Finite ι]
[Finite σ]
(P : PreSubmersivePresentation R S ι σ)
(T : Type u_5)
[CommRing T]
[Algebra S T]
[Algebra R T]
[IsScalarTower R S T]
[IsLocalization.Away P.jacobian T]
:
Inverting the selected Jacobian minor of a finite presentation gives a standard smooth algebra of the dimension of that presentation.