theorem FiniteMorphismDimension.PrimeSpectrum.comap_strictMono_of_isIntegral {B : Type u_1} {A : Type u_2} [CommRing B] [CommRing A] [Algebra B A] [Algebra.IsIntegral B A] : StrictMono (PrimeSpectrum.comap (algebraMap B A))

For an integral algebra extension B → A, the induced map on prime spectra is strictly monotone (incomparability of primes lying over the same prime).

theorem FiniteMorphismDimension.ringKrullDim_le_of_integral {B : Type u_1} {A : Type u_2} [CommRing B] [CommRing A] [Algebra B A] [Algebra.IsIntegral B A] : ringKrullDim A ≤ ringKrullDim B

For an integral extension B → A, the Krull dimension does not increase: dim A ≤ dim B.

theorem FiniteMorphismDimension.exists_lift_chain {B : Type u_1} {A : Type u_2} [CommRing B] [CommRing A] [Algebra B A] [Algebra.IsIntegral B A] (hinj : Function.Injective ⇑(algebraMap B A)) (n : ℕ) (f : Fin (n + 1) → PrimeSpectrum B) (hf : ∀ (i : Fin n), f i.castSucc < f i.succ) : ∃ (g : Fin (n + 1) → PrimeSpectrum A), (∀ (i : Fin (n + 1)), PrimeSpectrum.comap (algebraMap B A) (g i) = f i) ∧ ∀ (i : Fin n), g i.castSucc < g i.succ

Going-up: for an injective integral extension B → A, any strict chain of primes in B can be lifted to a strict chain of primes in A along the comap map.

theorem FiniteMorphismDimension.ringKrullDim_ge_of_injective_integral {B : Type u_1} {A : Type u_2} [CommRing B] [CommRing A] [Algebra B A] [Algebra.IsIntegral B A] (hinj : Function.Injective ⇑(algebraMap B A)) : ringKrullDim B ≤ ringKrullDim A

Consequence of going-up: an injective integral extension preserves Krull dimension from below, i.e. dim B ≤ dim A.

theorem FiniteMorphismDimension.ringKrullDim_eq_of_injective_finite {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)) : ringKrullDim A = ringKrullDim B

Lecture 5, Lemma 10 (ring-theoretic form): a finite injective extension preserves Krull dimension: dim A = dim B.

theorem ringKrullDim_eq_of_finite_injective {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] (φ : R →+* S) (hfin : Module.Finite R S) (hinj : Function.Injective ⇑φ) : ringKrullDim S = ringKrullDim R

Ring-hom phrasing of the dimension preservation result: a finite injective ring map preserves Krull dimension.

theorem mvPolynomial_Fin_maximal_height_eq_dim (k : Type u_1) [Field k] (n : ℕ) (m : Ideal (MvPolynomial (Fin n) k)) [m.IsMaximal] : ↑m.height = ringKrullDim (MvPolynomial (Fin n) k)

In a polynomial ring k[x_1, …, x_n] over a field, every maximal ideal has height equal to the Krull dimension n of the ring.

theorem ringKrullDim_mvPolynomial_quotient_non_constant (k : Type u_1) [Field k] (n : ℕ) (hn : n ≥ 1) (f : MvPolynomial (Fin n) k) (hf : f ∉ Set.range ⇑MvPolynomial.C) : ringKrullDim (MvPolynomial (Fin n) k ⧸ Ideal.span {f}) = ↑(n - 1)

Lecture 5, Corollary 10 / Theorem 5.4: the quotient of k[x_1, …, x_n] by a single non-constant polynomial has Krull dimension n - 1.