Theorems · Definition · commutative algebra

IsLocalization.AtPrime.equivQuotientMapMaximalIdeal

(S : Type u_6) →
  {R : Type u_7} →
    [inst : CommRing R] →
      (p : Ideal R) →
        [inst_1 : p.IsMaximal] →
          (Rₚ : Type u_8) →
            [inst_2 : CommRing Rₚ] →
              [inst_3 : Algebra R Rₚ] →
                [IsLocalization.AtPrime Rₚ p] →
                  [inst_5 : IsLocalRing Rₚ] →
                    (Sₚ : Type u_9) →
                      [inst_6 : CommRing S] →
                        [inst_7 : Algebra R S] →
                          [inst_8 : CommRing Sₚ] →
                            [inst_9 : Algebra S Sₚ] →
                              [inst_10 : Algebra R Sₚ] →
                                [inst_11 : Algebra Rₚ Sₚ] →
                                  [IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ] →
                                    [IsScalarTower R S Sₚ] →
                                      [IsScalarTower R Rₚ Sₚ] →
                                        S ⧸ Ideal.map (algebraMap R S) p ≃+*
                                          Sₚ ⧸ Ideal.map (algebraMap Rₚ Sₚ) (IsLocalRing.maximalIdeal Rₚ)

The isomorphism S ⧸ pS ≃+* Sₚ ⧸ p·Sₚ, where Sₚ is the localization of S at the (image) of the complement of p

Defined in

Mathlib.RingTheory.Localization.AtPrime.Basic

Cited by

2 results in Mathlib

Foundations

Depth 95 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingIdeal.IsMaximalCommRingAlgebraIsLocalization.AtPrimeIsLocalRingCommRingAlgebraCommRingAlgebraAlgebraAlgebraIsLocalizationIsScalarTowerIsScalarTower