Theorems · Theorem · commutative algebra

trace_quotient_eq_trace_localization_quotient

∀ {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.IsMaximal] (Rₚ : Type u_3) (Sₚ : Type u_4) [inst_4 : CommRing Rₚ] [inst_5 : CommRing Sₚ]
  [inst_6 : Algebra R Rₚ] [inst_7 : IsLocalization.AtPrime Rₚ p] [inst_8 : IsLocalRing Rₚ] [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] (x : S),
  (Algebra.trace (R ⧸ p) (S ⧸ Ideal.map (algebraMap R S) p)) ((Ideal.Quotient.mk (Ideal.map (algebraMap R S) p)) x) =
    (IsLocalization.AtPrime.equivQuotMaximalIdeal p Rₚ).symm
      ((Algebra.trace (Rₚ ⧸ IsLocalRing.maximalIdeal Rₚ)
          (Sₚ ⧸ Ideal.map (algebraMap Rₚ Sₚ) (IsLocalRing.maximalIdeal Rₚ)))
        ((algebraMap S (Sₚ ⧸ Ideal.map (algebraMap Rₚ Sₚ) (IsLocalRing.maximalIdeal Rₚ))) x))

Defined in

Mathlib.RingTheory.Trace.Quotient

Cited by

1 results in Mathlib

Foundations

Depth 110 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingCommRingAlgebraIdeal.IsMaximalCommRingCommRingAlgebraIsLocalization.AtPrimeIsLocalRingAlgebraAlgebraAlgebraIsLocalizationIsScalarTowerIsScalarTower