noncomputable def fiberKrullDim {B : Type u_1} {A : Type u_2} [CommRing B] [CommRing A] [Algebra B A] (q : PrimeSpectrum B) : WithBot ℕ∞

The Krull dimension of the (scheme-theoretic) fiber of Spec A → Spec B over a prime q ∈ Spec B, computed as the Krull dimension of A / qA.

def fiberDimGe {B : Type u_1} {A : Type u_2} [CommRing B] [CommRing A] [Algebra B A] (d : WithBot ℕ∞) : Set (PrimeSpectrum B)

The set of points q ∈ Spec B whose fiber in Spec A has Krull dimension at least d.

def fiberDimGeInImage {B : Type u_1} {A : Type u_2} [CommRing B] [CommRing A] [Algebra B A] (d : ℕ) : Set (PrimeSpectrum B)

The points in the image of Spec A → Spec B whose fiber has dimension at least d.

theorem fiberDimGeInImage_eq_inter {B : Type u_1} {A : Type u_2} [CommRing B] [CommRing A] [Algebra B A] (d : ℕ) : fiberDimGeInImage d = Set.range (PrimeSpectrum.comap (algebraMap B A)) ∩ fiberDimGe ↑d

The set of image points with fiber dimension ≥ d is the intersection of the image of Spec A → Spec B with the upper-level set fiberDimGe d.

theorem chevalley_thm21_2_part1 (B : Type u_1) (A : Type u_2) [CommRing B] [CommRing A] [IsNoetherianRing B] [Algebra B A] [Algebra.FiniteType B A] : Topology.IsConstructible (Set.range (PrimeSpectrum.comap (algebraMap B A)))

Chevalley's theorem (Theorem 21.2, part 1): For a finite-type morphism between Noetherian schemes, the image of Spec A → Spec B is a constructible subset.

theorem fiberDimGe_isClosed (B : Type u_1) (A : Type u_2) [CommRing B] [CommRing A] [IsDomain B] [IsDomain A] [IsNoetherianRing B] [Algebra B A] [Algebra.FiniteType B A] (hf : Function.Injective ⇑(algebraMap B A)) (d : ℕ) : IsClosed (fiberDimGe ↑d)

Upper-semicontinuity of fiber dimension: the locus of points in Spec B where the fiber of a dominant finite-type morphism has dimension at least d is closed.

theorem chevalley_thm21_2_part2 (B : Type u_1) (A : Type u_2) [CommRing B] [CommRing A] [IsDomain B] [IsDomain A] [IsNoetherianRing B] [Algebra B A] [Algebra.FiniteType B A] (hf : Function.Injective ⇑(algebraMap B A)) (d : ℕ) : ∃ (Z : Set (PrimeSpectrum B)), IsClosed Z ∧ fiberDimGeInImage d = Set.range (PrimeSpectrum.comap (algebraMap B A)) ∩ Z

Chevalley's theorem (Theorem 21.2, part 2): The locus in the image of a dominant finite-type morphism where the fiber has dimension at least d is closed in the image.

theorem chevalley_thm21_2_part3 (B : Type u_1) (A : Type u_2) [CommRing B] [CommRing A] [IsDomain B] [IsDomain A] [IsNoetherianRing B] [Algebra B A] [Algebra.FiniteType B A] (hf : Function.Injective ⇑(algebraMap B A)) : ∃ (b : B), b ≠ 0 ∧ ↑(PrimeSpectrum.basicOpen b) ⊆ Set.range (PrimeSpectrum.comap (algebraMap B A)) ∧ ∀ q ∈ ↑(PrimeSpectrum.basicOpen b), fiberKrullDim q + ringKrullDim B = ringKrullDim A

Chevalley's theorem (Theorem 21.2, part 3): For a dominant finite-type morphism, there is a nonempty basic open of the base on which the morphism is surjective and the dimension formula dim(fiber) + dim B = dim A holds.