Theorems · Definition · commutative algebra

IsDedekindDomain.primesOverEquivPrimesOver

{R : Type u_1} →
  {S : Type u_2} →
    [inst : CommRing R] →
      [inst_1 : CommRing S] →
        [inst_2 : Algebra R S] →
          (p : Ideal R) →
            [inst_3 : p.IsPrime] →
              (Rₚ : Type u_3) →
                [inst_4 : CommRing Rₚ] →
                  [inst_5 : Algebra R Rₚ] →
                    [IsLocalization.AtPrime Rₚ p] →
                      [inst_7 : IsLocalRing Rₚ] →
                        (Sₚ : Type u_4) →
                          [inst_8 : CommRing Sₚ] →
                            [inst_9 : Algebra S Sₚ] →
                              [IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ] →
                                [inst_11 : Algebra Rₚ Sₚ] →
                                  [IsDomain R] →
                                    [IsDedekindDomain S] →
                                      [Module.IsTorsionFree R S] →
                                        [inst_15 : Algebra R Sₚ] →
                                          [IsScalarTower R S Sₚ] →
                                            [IsScalarTower R Rₚ Sₚ] →
                                              p ≠ ⊥ →
                                                ↑(p.primesOver S) ≃o ↑((IsLocalRing.maximalIdeal Rₚ).primesOver Sₚ)

For R ⊆ S an extension of Dedekind domains and p a prime ideal of R, the bijection between the primes of S over p and the primes over the maximal ideal of Rₚ in Sₚ where Rₚ and Sₚ are resp. the localizations of R and S at the complement of p.

Defined in

Mathlib.RingTheory.Localization.AtPrime.Extension

Cited by

4 results in Mathlib

Foundations

Depth 138 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingCommRingAlgebraIdeal.IsPrimeCommRingAlgebraIsLocalization.AtPrimeIsLocalRingCommRingAlgebraIsLocalizationAlgebraIsDomainIsDedekindDomainModule.IsTorsionFreeAlgebraIsScalarTowerIsScalarTower