English
In a Dedekind domain, only finitely many height-one primes divide a given nonzero ideal.
Русский
В Дедекендовом домене только finitely многие высшие единицы делят данный ненулевой идеал.
LaTeX
$$$\{ v : HeightOneSpectrum R \mid v.asIdeal \mid I\}.\Finite$$$
Lean4
/-- For every nonzero ideal `I` of `v`, there are finitely many maximal ideals `v` such that the
multiplicity of `v` in the factorization of `I`, denoted `val_v(I)`, is nonzero. -/
theorem finite_factors {I : Ideal R} (hI : I ≠ 0) :
∀ᶠ v : HeightOneSpectrum R in Filter.cofinite,
((Associates.mk v.asIdeal).count (Associates.mk I).factors : ℤ) = 0 :=
by
have h_supp :
{v : HeightOneSpectrum R | ¬((Associates.mk v.asIdeal).count (Associates.mk I).factors : ℤ) = 0} =
{v : HeightOneSpectrum R | v.asIdeal ∣ I} :=
by
ext v
simp_rw [Int.natCast_eq_zero]
exact Associates.count_ne_zero_iff_dvd hI v.irreducible
rw [Filter.eventually_cofinite, h_supp]
exact Ideal.finite_factors hI
