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abbrev Lec4.IsAffineMorphism {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) : Prop

Lecture 4, Definition 9: affine morphism of schemes, defined here in terms of Mathlib's IsAffineHom.

@[reducible, inline]

abbrev Lec4.IsFiniteMorphism {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) : Prop

Lecture 4, Definition 10: a finite morphism of schemes, defined here as Mathlib's IsFinite f.

theorem Lec4.finiteMorphism_closedMap {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) [AlgebraicGeometry.IsFinite f] : IsClosedMap ⇑f

Lecture 4, Lemma 7 (closedness part): a finite morphism of schemes is a closed map.

theorem Lec4.finiteMorphism_finitePreimage {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) [AlgebraicGeometry.IsFinite f] (y : ↥Y) : (⇑f ⁻¹' {y}).Finite

Lecture 4, Lemma 7 (finite-fibers part): a finite morphism has finite fibers.

theorem Lec4.comap_surjective_of_finite_injective {B : Type u_1} {A : Type u_2} [CommRing B] [CommRing A] [Algebra B A] [Module.Finite B A] (hinj : Function.Injective ⇑(algebraMap B A)) : Function.Surjective (PrimeSpectrum.comap (algebraMap B A))

Ring-theoretic version of Lecture 4, Corollary 9: if A is a finite injective B-algebra, then Spec A → Spec B is surjective.

theorem Lec4.comap_finite_fibers_of_finite {B : Type u_1} {A : Type u_2} [CommRing B] [CommRing A] [Algebra B A] [Module.Finite B A] (p : PrimeSpectrum B) : (PrimeSpectrum.comap (algebraMap B A) ⁻¹' {p}).Finite

Ring-theoretic version of finite fibers: for a finite B-algebra A, the map Spec A → Spec B has finite fibers.

theorem Lec4.comap_lt_comap_of_finite {B : Type u_1} {A : Type u_2} [CommRing B] [CommRing A] [Algebra B A] [Module.Finite B A] {I J : Ideal A} [I.IsPrime] (hIJ : I < J) : Ideal.comap (algebraMap B A) I < Ideal.comap (algebraMap B A) J

"Incomparability" for finite extensions: for A finite over B, a strict containment of primes I < J in A contracts to a strict containment in B.

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noncomputable abbrev Lec4.krullDim_def (T : Type u_1) [TopologicalSpace T] : WithBot ℕ∞

Convenience wrapper: the topological Krull dimension of T, used here as the Lec 4 notion of dimension.