Theorems · Theorem · commutative algebra
Localization.AtPrime.under_maximalIdeal
∀ {R : Type u_1} [inst : CommSemiring R] {I : Ideal R} [hI : I.IsPrime],
Ideal.under R (IsLocalRing.maximalIdeal (Localization I.primeCompl)) = IThe unique maximal ideal of the localization at I.primeCompl lies over the ideal I.
- Cited by
- 9 results in Mathlib
- Foundations
- Depth 86 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommSemiringIdeal.IsPrime
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites9
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- CommSemiringstatement and proof · cited by 10,911
- Idealstatement and proof · cited by 4,748
- Ideal.IsPrimestatement and proof · cited by 827
- Ideal.primeComplstatement and proof · cited by 462
- IsLocalRing.maximalIdealstatement · cited by 297
- Localizationstatement and proof · cited by 270
- Ideal.understatement · cited by 170
- IsLocalization.AtPrime.isLocalRingproof · cited by 19
- IsLocalization.AtPrime.under_maximalIdealproof · cited by 7
Cited by9
Results whose statement or proof uses this declaration.
- Localization.AtPrime.map_eq_maximalIdealproof · cited by 17
- Ideal.height_le_spanRank_toENat_of_mem_minimalPrimesproof · cited by 5
- Ideal.height_le_one_of_isPrincipal_of_mem_minimalPrimesproof · cited by 4
- Ideal.exists_ideal_over_prime_of_isIntegral_of_isDomainproof · cited by 2
- Algebra.QuasiFiniteAt.eq_of_le_of_under_eqproof · cited by 1
- AlgHom.IsArithFrobAt.isArithFrobAt_localizeproof · cited by 1
- Localization.AtPrime.eq_maximalIdeal_iff_under_eqproof · cited by 1
- Localization.AtPrime.comap_maximalIdealproof · cited by 0