theorem not_le_heightOnePrime_of_height_ge_two {A : Type u_1} [CommRing A] (I : Ideal A) (hI : 2 ≤ I.height) (p : AlgebraicHartogs.HeightOnePrime A) : ¬I ≤ p.asIdeal

An ideal of height at least 2 cannot be contained in a height-one prime.

theorem Proposition39.denominator_ideal_self_improvement {A : Type u_3} [CommRing A] [IsDomain A] [IsNoetherianRing A] [IsIntegrallyClosed A] {K : Type u_4} [Field K] [Algebra A K] [IsFractionRing A K] (x : K) (hx : ∀ (p : AlgebraicHartogs.HeightOnePrime A), x ∈ Localization.subalgebra.ofField K p.asIdeal.primeCompl ⋯) (c : A) (hc : (algebraMap A K) c * x ∈ Set.range ⇑(algebraMap A K)) : (algebraMap A K) c * x * x ∈ Set.range ⇑(algebraMap A K)

Key technical step: if x ∈ K is locally regular at every height-one prime and c · x already lies in A, then c · x² also lies in A. Used inductively to bound denominators when proving integrality.

noncomputable def Proposition39.divByMap {A : Type u_1} [CommRing A] {K : Type u_2} [Field K] [Algebra A K] (b : A) : A →ₗ[A] K

The A-linear map A → K sending a ↦ a / b, used to realise A[x] ⊆ K as a submodule of a finitely generated A-module when b clears all denominators of A[x].

theorem Proposition39.isIntegral_of_mem_heightOnePrime_localizations {A : Type u_1} [CommRing A] [IsDomain A] [IsNoetherianRing A] [IsIntegrallyClosed A] {K : Type u_2} [Field K] [Algebra A K] [IsFractionRing A K] (x : K) (hx : ∀ (p : AlgebraicHartogs.HeightOnePrime A), x ∈ Localization.subalgebra.ofField K p.asIdeal.primeCompl ⋯) : IsIntegral A x

If x ∈ K lies in the localization of A at every height-one prime, then x is integral over A. This is a key input to Hartogs/normal-domain results.

theorem Proposition39.iInf_heightOnePrime_localization_eq_bot {A : Type u_1} [CommRing A] [IsDomain A] [IsNoetherianRing A] [IsIntegrallyClosed A] {K : Type u_2} [Field K] [Algebra A K] [IsFractionRing A K] : ⨅ (p : AlgebraicHartogs.HeightOnePrime A), Localization.subalgebra.ofField K p.asIdeal.primeCompl ⋯ = ⊥

For an integrally closed Noetherian domain, the intersection of all height-one localizations inside its fraction field equals A itself.

theorem Proposition39.mem_of_mem_heightOnePrime_localizations {A : Type u_1} [CommRing A] [IsDomain A] [IsNoetherianRing A] [IsIntegrallyClosed A] {K : Type u_2} [Field K] [Algebra A K] [IsFractionRing A K] (x : K) (hx : ∀ (p : AlgebraicHartogs.HeightOnePrime A), x ∈ Localization.subalgebra.ofField K p.asIdeal.primeCompl ⋯) : x ∈ Set.range ⇑(algebraMap A K)

An element of the fraction field that lies in every height-one localization of a normal Noetherian domain already lies in the ring itself.

theorem Proposition39.proposition39_codim2_extension {A : Type u_1} [CommRing A] [IsDomain A] [IsNoetherianRing A] [IsIntegrallyClosed A] {K : Type u_2} [Field K] [Algebra A K] [IsFractionRing A K] (I : Ideal A) (hI : 2 ≤ I.height) (x : K) (hx : ∀ (p : PrimeSpectrum A), ¬I ≤ p.asIdeal → x ∈ Localization.subalgebra.ofField K p.asIdeal.primeCompl ⋯) : x ∈ Set.range ⇑(algebraMap A K)

Proposition 39 (codimension-2 extension): For a normal Noetherian domain, any element of the fraction field that is regular outside a closed subset of codimension ≥ 2 extends globally to a regular function on Spec A.

theorem Proposition39.proposition39_range {A : Type u_1} [CommRing A] [IsDomain A] [IsNoetherianRing A] [IsIntegrallyClosed A] {K : Type u_2} [Field K] [Algebra A K] [IsFractionRing A K] (I : Ideal A) (hI : 2 ≤ I.height) : Set.range ⇑(algebraMap A K) = ⋂ (p : PrimeSpectrum A), ⋂ (_ : ¬I ≤ p.asIdeal), ↑(Localization.subalgebra.ofField K p.asIdeal.primeCompl ⋯)

Proposition 39 (set-theoretic form): For a normal Noetherian domain A and an ideal of height at least 2, A equals the intersection of all localizations at primes not containing that ideal.