theorem normal_codim2_extension (R : Type u_1) [CommRing R] [IsDomain R] [IsNoetherianRing R] [IsIntegrallyClosed R] (K : Type u_2) [Field K] [Algebra R K] [IsFractionRing R K] (I : Ideal R) (hI : 2 ≤ I.height) : ⨅ (p : PrimeSpectrum R), ⨅ (_ : ¬I ≤ p.asIdeal), Localization.subalgebra.ofField K p.asIdeal.primeCompl ⋯ = ⊥

Proposition 39 (Lecture 20). On a normal Noetherian domain R, the intersection over primes p with I ⊄ p of the localizations R_p ⊂ K is trivial (equal to R itself, i.e. the bottom subalgebra), provided I has height ≥ 2. This is the algebraic form of "regular functions extend across closed subsets of codimension ≥ 2 in a normal variety".

theorem normal_codim2_extension_mem (R : Type u_1) [CommRing R] [IsDomain R] [IsNoetherianRing R] [IsIntegrallyClosed R] (K : Type u_2) [Field K] [Algebra R K] [IsFractionRing R K] (I : Ideal R) (hI : 2 ≤ I.height) (x : K) (hx : ∀ (p : PrimeSpectrum R), ¬I ≤ p.asIdeal → x ∈ Localization.subalgebra.ofField K p.asIdeal.primeCompl ⋯) : x ∈ Set.range ⇑(algebraMap R K)

Element-level form of Proposition 39: an element of the fraction field K that lies in R_p for every prime p outside the codimension-≥-2 locus V(I) is in fact in the image of R → K.