noncomputable def dimNat (R : Type u_1) [CommRing R] : ℕ

Krull dimension of a commutative ring as a natural number (returning 0 in degenerate or infinite cases).

theorem equidimensional_mvpoly_nat (k : Type u_1) [Field k] (n : ℕ) (Q : Ideal (MvPolynomial (Fin n) k)) [Q.IsPrime] : Q.height.toNat + dimNat (MvPolynomial (Fin n) k ⧸ Q) = n

Equidimensionality of polynomial rings (Lec 8, Prop 10): for any prime ideal Q in k[x_1,…,x_n], height Q plus dimension of the quotient equals n.

theorem intersection_dim_bound_nat (k : Type u_1) [Field k] (n : ℕ) (I J : Ideal (MvPolynomial (Fin n) k)) [I.IsPrime] [J.IsPrime] (P : Ideal (MvPolynomial (Fin n) k)) (hP : P ∈ (I ⊔ J).minimalPrimes) : dimNat (MvPolynomial (Fin n) k ⧸ I) + dimNat (MvPolynomial (Fin n) k ⧸ J) ≤ dimNat (MvPolynomial (Fin n) k ⧸ P) + n

Lec 8, Thm 8.1 (intersection codimension, dimension form): for any two irreducible subvarieties of 𝔸ⁿ and a minimal prime P over their sum, dim X + dim Y ≤ dim(X ∩ Y) + n.

theorem thm81_intersection_codim_bound (k : Type u_1) [Field k] (n : ℕ) (I J : Ideal (MvPolynomial (Fin n) k)) [I.IsPrime] [J.IsPrime] (P : Ideal (MvPolynomial (Fin n) k)) (hP : P ∈ (I ⊔ J).minimalPrimes) : P.height ≤ I.height + J.height

Lec 8, Thm 8.1 (intersection codimension): for prime ideals I, J ⊆ k[x_1,…,x_n] and any minimal prime P over I + J, the codimension of P is at most the sum of codimensions of I, J.

theorem height_minPrime_sup_le_spanRank_add {R : Type u_1} [CommRing R] [IsNoetherianRing R] (I J P : Ideal R) (hP : P ∈ (I ⊔ J).minimalPrimes) : P.height ≤ Cardinal.toENat (Submodule.spanRank I) + Cardinal.toENat (Submodule.spanRank J)

A minimal prime over I + J in a noetherian ring has height at most the sum of the span-ranks of I and J (via Krull's height theorem).

theorem height_minPrime_sup_le_spanFinrank_add {R : Type u_1} [CommRing R] [IsNoetherianRing R] (I J P : Ideal R) (hP : P ∈ (I ⊔ J).minimalPrimes) : P.height ≤ ↑(Submodule.spanFinrank I + Submodule.spanFinrank J)

A minimal prime over I + J in a noetherian ring has height at most the sum of the (finite) span-ranks of I and J.