Theorem: Ideal.spanIntNorm_localization
Proof state before:
R : Type u_1
inst✝²⁵ : CommRing R
S : Type u_2
inst✝²⁴ : CommRing S
inst✝²³ : Algebra R S
inst✝²² : IsIntegrallyClosed R
inst✝²¹ : IsDomain R
inst✝²⁰ : IsDomain S
inst✝¹⁹ : NoZeroSMulDivisors R S
hRS : Module.Finite R S
inst✝¹⁸ : IsIntegrallyClosed S
inst✝¹⁷ : IsSeparable (FractionRing R) (FractionRing S)
I : Ideal S
M : Submonoid R
hM : M ≤ R⁰
Rₘ : Type u_3
Sₘ : Type u_4
inst✝¹⁶ : CommRing Rₘ
inst✝¹⁵ : Algebra R Rₘ
inst✝¹⁴ : CommRing Sₘ
inst✝¹³ : Algebra S Sₘ
inst✝¹² : Algebra Rₘ Sₘ
inst✝¹¹ : Algebra R Sₘ
inst✝¹⁰ : IsScalarTower R Rₘ Sₘ
inst✝⁹ : IsScalarTower R S Sₘ
inst✝⁸ : IsLocalization M Rₘ
inst✝⁷ : IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ
inst✝⁶ : IsIntegrallyClosed Rₘ
inst✝⁵ : IsDomain Rₘ
inst✝⁴ : IsDomain Sₘ
inst✝³ : NoZeroSMulDivisors Rₘ Sₘ
inst✝² : Module.Finite Rₘ Sₘ
inst✝¹ : IsIntegrallyClosed Sₘ
inst✝ : IsSeparable (FractionRing Rₘ) (FractionRing Sₘ)
K : Type u_1 := FractionRing R
f : Rₘ →+* K := IsLocalization.map K (RingHom.id R) hM
this✝⁵ : Algebra Rₘ K := f.toAlgebra
this✝⁴ : IsScalarTower R Rₘ K
this✝³ : IsFractionRing Rₘ K := IsFractionRing.isFractionRing_of_isDomain_of_isLocalization M Rₘ K
L : Type u_2 := FractionRing S
g : Sₘ →+* L := IsLocalization.map L (RingHom.id S) ⋯
this✝² : Algebra Sₘ L := g.toAlgebra
this✝¹ : IsScalarTower S Sₘ L
this✝ : Algebra Rₘ L := ((algebraMap K L).comp f).toAlgebra
this : IsScalarTower Rₘ K L
⊢ spanIntNorm Rₘ (map (algebraMap S Sₘ) I) = map (algebraMap R Rₘ) (spanIntNorm R I)
Tactic:
haveI : IsScalarTower Rₘ Sₘ L := by
  apply IsScalarTower.of_algebraMap_eq'
  apply IsLocalization.ringHom_ext M
  rw [RingHom.algebraMap_toAlgebra, RingHom.algebraMap_toAlgebra (R := Sₘ), RingHom.comp_assoc,
    RingHom.comp_assoc, ← IsScalarTower.algebraMap_eq, IsScalarTower.algebraMap_eq R S Sₘ,
    IsLocalization.map_comp, RingHom.comp_id, ← RingHom.comp_assoc, IsLocalization.map_comp,
    RingHom.comp_id, ← IsScalarTower.algebraMap_eq, ← IsScalarTower.algebraMap_eq]
Proof state after:
R : Type u_1
inst✝²⁵ : CommRing R
S : Type u_2
inst✝²⁴ : CommRing S
inst✝²³ : Algebra R S
inst✝²² : IsIntegrallyClosed R
inst✝²¹ : IsDomain R
inst✝²⁰ : IsDomain S
inst✝¹⁹ : NoZeroSMulDivisors R S
hRS : Module.Finite R S
inst✝¹⁸ : IsIntegrallyClosed S
inst✝¹⁷ : IsSeparable (FractionRing R) (FractionRing S)
I : Ideal S
M : Submonoid R
hM : M ≤ R⁰
Rₘ : Type u_3
Sₘ : Type u_4
inst✝¹⁶ : CommRing Rₘ
inst✝¹⁵ : Algebra R Rₘ
inst✝¹⁴ : CommRing Sₘ
inst✝¹³ : Algebra S Sₘ
inst✝¹² : Algebra Rₘ Sₘ
inst✝¹¹ : Algebra R Sₘ
inst✝¹⁰ : IsScalarTower R Rₘ Sₘ
inst✝⁹ : IsScalarTower R S Sₘ
inst✝⁸ : IsLocalization M Rₘ
inst✝⁷ : IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ
inst✝⁶ : IsIntegrallyClosed Rₘ
inst✝⁵ : IsDomain Rₘ
inst✝⁴ : IsDomain Sₘ
inst✝³ : NoZeroSMulDivisors Rₘ Sₘ
inst✝² : Module.Finite Rₘ Sₘ
inst✝¹ : IsIntegrallyClosed Sₘ
inst✝ : IsSeparable (FractionRing Rₘ) (FractionRing Sₘ)
K : Type u_1 := FractionRing R
f : Rₘ →+* K := IsLocalization.map K (RingHom.id R) hM
this✝⁶ : Algebra Rₘ K := f.toAlgebra
this✝⁵ : IsScalarTower R Rₘ K
this✝⁴ : IsFractionRing Rₘ K := IsFractionRing.isFractionRing_of_isDomain_of_isLocalization M Rₘ K
L : Type u_2 := FractionRing S
g : Sₘ →+* L := IsLocalization.map L (RingHom.id S) ⋯
this✝³ : Algebra Sₘ L := g.toAlgebra
this✝² : IsScalarTower S Sₘ L
this✝¹ : Algebra Rₘ L := ((algebraMap K L).comp f).toAlgebra
this✝ : IsScalarTower Rₘ K L
this : IsScalarTower Rₘ Sₘ L
⊢ spanIntNorm Rₘ (map (algebraMap S Sₘ) I) = map (algebraMap R Rₘ) (spanIntNorm R I)