class
Formalization.Lec7CompleteVariety.IsCompleteVariety
{X S : AlgebraicGeometry.Scheme}
(f : X ⟶ S)
:
A morphism f : X ⟶ S is a complete variety if it is separated
and universally closed (Lec 7/8, Def 19 / Lem 19 setting).
- isSeparated : AlgebraicGeometry.IsSeparated f
- universallyClosed : AlgebraicGeometry.UniversallyClosed f
Instances
instance
Formalization.Lec7CompleteVariety.isCompleteVariety_of_isProper
{X S : AlgebraicGeometry.Scheme}
(f : X ⟶ S)
[AlgebraicGeometry.IsProper f]
:
A proper morphism is a complete variety.
theorem
Formalization.Lec7CompleteVariety.isProper_of_isCompleteVariety
{X S : AlgebraicGeometry.Scheme}
(f : X ⟶ S)
[hc : IsCompleteVariety f]
[AlgebraicGeometry.LocallyOfFiniteType f]
:
A complete variety that is locally of finite type is proper.