theorem isClosedImmersion_iff_closedRange_embedding_surjectiveOnStalks {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) : AlgebraicGeometry.IsClosedImmersion f ↔ IsClosed (Set.range ⇑f) ∧ Topology.IsEmbedding ⇑f ∧ ∀ (x : ↥X), Function.Surjective ⇑(CategoryTheory.ConcreteCategory.hom (AlgebraicGeometry.Scheme.Hom.stalkMap f x))

Characterisation of closed immersions of schemes: f is a closed immersion iff its image is closed, the underlying map is a topological embedding, and every stalk map is surjective.

theorem IsClosedImmersion.isClosed_range' {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) [AlgebraicGeometry.IsClosedImmersion f] : IsClosed (Set.range ⇑f)

The image of a closed immersion is a closed subset of the target.

theorem IsClosedImmersion.isEmbedding' {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) [AlgebraicGeometry.IsClosedImmersion f] : Topology.IsEmbedding ⇑f

The underlying continuous map of a closed immersion is a topological embedding.

theorem IsClosedImmersion.stalkMap_surjective' {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) [AlgebraicGeometry.IsClosedImmersion f] (x : ↥X) : Function.Surjective ⇑(CategoryTheory.ConcreteCategory.hom (AlgebraicGeometry.Scheme.Hom.stalkMap f x))

The stalk map of a closed immersion at any point is surjective.