Quasi-separated morphisms #
A morphism of schemes f : X ⟶ Y
is quasi-separated if the diagonal morphism X ⟶ X ×[Y] X
is
quasi-compact.
A scheme is quasi-separated if the intersections of any two affine open sets is quasi-compact.
(AlgebraicGeometry.quasiSeparatedSpace_iff_affine
)
We show that a morphism is quasi-separated if the preimage of every affine open is quasi-separated.
We also show that this property is local at the target, and is stable under compositions and base-changes.
Main result #
AlgebraicGeometry.is_localization_basicOpen_of_qcqs
(Qcqs lemma): IfU
is qcqs, thenΓ(X, D(f)) ≃ Γ(X, U)_f
for everyf : Γ(X, U)
.
A morphism is QuasiSeparated
if diagonal map is quasi-compact.
- diagonalQuasiCompact : AlgebraicGeometry.QuasiCompact (CategoryTheory.Limits.pullback.diagonal f)
A morphism is
QuasiSeparated
if diagonal map is quasi-compact.
Instances
A morphism is QuasiSeparated
if diagonal map is quasi-compact.
Equations
- ⋯ = ⋯
Equations
- ⋯ = ⋯
Equations
- ⋯ = ⋯
Equations
- ⋯ = ⋯
Equations
- ⋯ = ⋯
If U
is qcqs, then Γ(X, D(f)) ≃ Γ(X, U)_f
for every f : Γ(X, U)
.
This is known as the Qcqs lemma in R. Vakil, The rising sea.
If U
is qcqs, then Γ(X, D(f)) ≃ Γ(X, U)_f
for every f : Γ(X, U)
.
This is known as the Qcqs lemma in R. Vakil, The rising sea.