Theorems · Theorem · commutative algebra
MaximalSpectrum.iInf_localization_eq_bot
∀ (R : Type u_4) [inst : CommRing R] [inst_1 : IsDomain R] (K : Type u_5) [inst_2 : Field K] [inst_3 : Algebra R K] [inst_4 : IsFractionRing R K], ⨅ v, Localization.subalgebra.ofField K v.asIdeal.primeCompl ⋯ = ⊥
An integral domain is equal to the intersection of its localizations at all its maximal ideals viewed as subalgebras of its field of fractions.
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 79 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites38
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- CommRingstatement and proof · cited by 17,173
- Algebrastatement and proof · cited by 11,388
- Fieldstatement and proof · cited by 7,404
- Idealproof · cited by 4,748
- Bot.botstatement · cited by 4,720
- Algebra.algebraMapproof · cited by 4,706
- Set.rangeproof · cited by 4,705
- mul_oneproof · cited by 3,885
- IsDomainstatement and proof · cited by 2,196
- iInfstatement and proof · cited by 1,690
- one_smulproof · cited by 1,374
Cited by3
Results whose statement or proof uses this declaration.
- IsIntegrallyClosed.of_localization_maximalproof · cited by 2
- IsDedekindDomain.HeightOneSpectrum.iInf_localization_eq_botproof · cited by 0
- PrimeSpectrum.iInf_localization_eq_botproof · cited by 0