theorem
Lec7.separated_of_open_immersion_to_separated
{X Y : AlgebraicGeometry.Scheme}
(f : X ⟶ Y)
[AlgebraicGeometry.IsOpenImmersion f]
[Y.IsSeparated]
:
Lec 7, Lem 17 / Cor 12: an open subscheme of a separated scheme is separated.
theorem
Lec7.prop9_forward
{X : AlgebraicGeometry.Scheme}
[X.IsSeparated]
{U V : X.Opens}
(hU : AlgebraicGeometry.IsAffineOpen U)
(hV : AlgebraicGeometry.IsAffineOpen V)
:
AlgebraicGeometry.IsAffineOpen (U ⊓ V)
Lec 7, Prop 9 (forward direction): on a separated scheme, the intersection of two affine opens is affine.
theorem
Lec7.prop9_separated_iff
{X : AlgebraicGeometry.Scheme}
:
X.IsSeparated ↔ (∀ (U V : X.Opens),
AlgebraicGeometry.IsAffineOpen U → AlgebraicGeometry.IsAffineOpen V → AlgebraicGeometry.IsAffineOpen (U ⊓ V)) ∧ ∀
(W :
(CategoryTheory.Limits.pullback (CategoryTheory.Limits.terminal.from X)
(CategoryTheory.Limits.terminal.from X)).Opens),
AlgebraicGeometry.IsAffineOpen W →
Function.Surjective
⇑(CategoryTheory.ConcreteCategory.hom
(AlgebraicGeometry.Scheme.Hom.app
(CategoryTheory.Limits.pullback.diagonal (CategoryTheory.Limits.terminal.from X)) W))
Lec 7, Prop 9: characterization of separatedness via affine intersections plus surjectivity of the diagonal on affine opens of the fiber product.
theorem
Lec7.proj_is_separated
{A σ : Type u}
[CommRing A]
[SetLike σ A]
[AddSubgroupClass σ A]
(𝒜 : ℕ → σ)
[GradedRing 𝒜]
:
Lec 7, Lem 18: Proj 𝒜 (in particular ℙⁿ) is separated.
theorem
Lec7.separated_of_closed_immersion_to_separated
{X Y : AlgebraicGeometry.Scheme}
(f : X ⟶ Y)
[AlgebraicGeometry.IsClosedImmersion f]
[Y.IsSeparated]
:
Lec 7: a closed subscheme of a separated scheme is separated.
theorem
Lec7.separated_of_locally_closed_immersion
{X Z Y : AlgebraicGeometry.Scheme}
(f : X ⟶ Z)
(g : Z ⟶ Y)
[AlgebraicGeometry.IsClosedImmersion f]
[AlgebraicGeometry.IsOpenImmersion g]
[Y.IsSeparated]
:
Lec 7, Lem 17: a locally closed subscheme of a separated scheme is separated.