Theorem: trace_quotient_eq_trace_localization_quotient (FltRegular/NumberTheory/QuotientTrace.lean)

State before:
R : Type u_1
S : Type u_2
inst✝¹⁵ : CommRing R
inst✝¹⁴ : CommRing S
inst✝¹³ : Algebra R S
p : Ideal R
inst✝¹² : p.IsMaximal
Rₚ : Type u_3
Sₚ : Type u_4
inst✝¹¹ : CommRing Rₚ
inst✝¹⁰ : CommRing Sₚ
inst✝⁹ : Algebra R Rₚ
inst✝⁸ : IsLocalization.AtPrime Rₚ p
inst✝⁷ : LocalRing Rₚ
inst✝⁶ : CommRing Sₚ
inst✝⁵ : Algebra S Sₚ
inst✝⁴ : Algebra R Sₚ
inst✝³ : Algebra Rₚ Sₚ
inst✝² : IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ
inst✝¹ : IsScalarTower R S Sₚ
inst✝ : IsScalarTower R Rₚ Sₚ
x : S
⊢ (Algebra.trace (R ⧸ p) (S ⧸ pS)) ((Ideal.Quotient.mk pS) x) =
    (equivQuotMaximalIdealOfIsLocalization p Rₚ).symm
      ((Algebra.trace (Rₚ ⧸ maximalIdeal Rₚ) (Sₚ ⧸ pSₚ)) ((algebraMap S (Sₚ ⧸ pSₚ)) x))
Tactic:
have : IsScalarTower R (Rₚ ⧸ maximalIdeal Rₚ) (Sₚ ⧸ pSₚ) := by
  apply IsScalarTower.of_algebraMap_eq'
  rw [IsScalarTower.algebraMap_eq R Rₚ (Rₚ ⧸ _), IsScalarTower.algebraMap_eq R Rₚ (Sₚ ⧸ _),
    ← RingHom.comp_assoc, ← IsScalarTower.algebraMap_eq Rₚ]
State after:
R : Type u_1
S : Type u_2
inst✝¹⁵ : CommRing R
inst✝¹⁴ : CommRing S
inst✝¹³ : Algebra R S
p : Ideal R
inst✝¹² : p.IsMaximal
Rₚ : Type u_3
Sₚ : Type u_4
inst✝¹¹ : CommRing Rₚ
inst✝¹⁰ : CommRing Sₚ
inst✝⁹ : Algebra R Rₚ
inst✝⁸ : IsLocalization.AtPrime Rₚ p
inst✝⁷ : LocalRing Rₚ
inst✝⁶ : CommRing Sₚ
inst✝⁵ : Algebra S Sₚ
inst✝⁴ : Algebra R Sₚ
inst✝³ : Algebra Rₚ Sₚ
inst✝² : IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ
inst✝¹ : IsScalarTower R S Sₚ
inst✝ : IsScalarTower R Rₚ Sₚ
x : S
this : IsScalarTower R (Rₚ ⧸ maximalIdeal Rₚ) (Sₚ ⧸ pSₚ)
⊢ (Algebra.trace (R ⧸ p) (S ⧸ pS)) ((Ideal.Quotient.mk pS) x) =
    (equivQuotMaximalIdealOfIsLocalization p Rₚ).symm
      ((Algebra.trace (Rₚ ⧸ maximalIdeal Rₚ) (Sₚ ⧸ pSₚ)) ((algebraMap S (Sₚ ⧸ pSₚ)) x))

State before:
R : Type u_1
S : Type u_2
inst✝¹⁵ : CommRing R
inst✝¹⁴ : CommRing S
inst✝¹³ : Algebra R S
p : Ideal R
inst✝¹² : p.IsMaximal
Rₚ : Type u_3
Sₚ : Type u_4
inst✝¹¹ : CommRing Rₚ
inst✝¹⁰ : CommRing Sₚ
inst✝⁹ : Algebra R Rₚ
inst✝⁸ : IsLocalization.AtPrime Rₚ p
inst✝⁷ : LocalRing Rₚ
inst✝⁶ : CommRing Sₚ
inst✝⁵ : Algebra S Sₚ
inst✝⁴ : Algebra R Sₚ
inst✝³ : Algebra Rₚ Sₚ
inst✝² : IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ
inst✝¹ : IsScalarTower R S Sₚ
inst✝ : IsScalarTower R Rₚ Sₚ
x : S
this : IsScalarTower R (Rₚ ⧸ maximalIdeal Rₚ) (Sₚ ⧸ pSₚ)
⊢ (Algebra.trace (R ⧸ p) (S ⧸ pS)) ((Ideal.Quotient.mk pS) x) =
    (equivQuotMaximalIdealOfIsLocalization p Rₚ).symm
      ((Algebra.trace (Rₚ ⧸ maximalIdeal Rₚ) (Sₚ ⧸ pSₚ)) ((algebraMap S (Sₚ ⧸ pSₚ)) x))
Tactic:
rw [Algebra.trace_eq_of_equiv_equiv (equivQuotMaximalIdealOfIsLocalization p Rₚ)
  (quotMapEquivQuotMapMaximalIdealOfIsLocalization S p Rₚ Sₚ)]
State after:
R : Type u_1
S : Type u_2
inst✝¹⁵ : CommRing R
inst✝¹⁴ : CommRing S
inst✝¹³ : Algebra R S
p : Ideal R
inst✝¹² : p.IsMaximal
Rₚ : Type u_3
Sₚ : Type u_4
inst✝¹¹ : CommRing Rₚ
inst✝¹⁰ : CommRing Sₚ
inst✝⁹ : Algebra R Rₚ
inst✝⁸ : IsLocalization.AtPrime Rₚ p
inst✝⁷ : LocalRing Rₚ
inst✝⁶ : CommRing Sₚ
inst✝⁵ : Algebra S Sₚ
inst✝⁴ : Algebra R Sₚ
inst✝³ : Algebra Rₚ Sₚ
inst✝² : IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ
inst✝¹ : IsScalarTower R S Sₚ
inst✝ : IsScalarTower R Rₚ Sₚ
x : S
this : IsScalarTower R (Rₚ ⧸ maximalIdeal Rₚ) (Sₚ ⧸ pSₚ)
⊢ (equivQuotMaximalIdealOfIsLocalization p Rₚ).symm
      ((Algebra.trace (Rₚ ⧸ maximalIdeal Rₚ) (Sₚ ⧸ pSₚ))
        ((quotMapEquivQuotMapMaximalIdealOfIsLocalization S p Rₚ Sₚ) ((Ideal.Quotient.mk pS) x))) =
    (equivQuotMaximalIdealOfIsLocalization p Rₚ).symm
      ((Algebra.trace (Rₚ ⧸ maximalIdeal Rₚ) (Sₚ ⧸ pSₚ)) ((algebraMap S (Sₚ ⧸ pSₚ)) x))

case he
R : Type u_1
S : Type u_2
inst✝¹⁵ : CommRing R
inst✝¹⁴ : CommRing S
inst✝¹³ : Algebra R S
p : Ideal R
inst✝¹² : p.IsMaximal
Rₚ : Type u_3
Sₚ : Type u_4
inst✝¹¹ : CommRing Rₚ
inst✝¹⁰ : CommRing Sₚ
inst✝⁹ : Algebra R Rₚ
inst✝⁸ : IsLocalization.AtPrime Rₚ p
inst✝⁷ : LocalRing Rₚ
inst✝⁶ : CommRing Sₚ
inst✝⁵ : Algebra S Sₚ
inst✝⁴ : Algebra R Sₚ
inst✝³ : Algebra Rₚ Sₚ
inst✝² : IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ
inst✝¹ : IsScalarTower R S Sₚ
inst✝ : IsScalarTower R Rₚ Sₚ
x : S
this : IsScalarTower R (Rₚ ⧸ maximalIdeal Rₚ) (Sₚ ⧸ pSₚ)
⊢ (algebraMap (Rₚ ⧸ maximalIdeal Rₚ) (Sₚ ⧸ pSₚ)).comp ↑(equivQuotMaximalIdealOfIsLocalization p Rₚ) =
    (↑(quotMapEquivQuotMapMaximalIdealOfIsLocalization S p Rₚ Sₚ)).comp (algebraMap (R ⧸ p) (S ⧸ pS))

State before:
R : Type u_1
S : Type u_2
inst✝¹⁵ : CommRing R
inst✝¹⁴ : CommRing S
inst✝¹³ : Algebra R S
p : Ideal R
inst✝¹² : p.IsMaximal
Rₚ : Type u_3
Sₚ : Type u_4
inst✝¹¹ : CommRing Rₚ
inst✝¹⁰ : CommRing Sₚ
inst✝⁹ : Algebra R Rₚ
inst✝⁸ : IsLocalization.AtPrime Rₚ p
inst✝⁷ : LocalRing Rₚ
inst✝⁶ : CommRing Sₚ
inst✝⁵ : Algebra S Sₚ
inst✝⁴ : Algebra R Sₚ
inst✝³ : Algebra Rₚ Sₚ
inst✝² : IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ
inst✝¹ : IsScalarTower R S Sₚ
inst✝ : IsScalarTower R Rₚ Sₚ
x : S
⊢ IsScalarTower R (Rₚ ⧸ maximalIdeal Rₚ) (Sₚ ⧸ pSₚ)
Tactic:
apply IsScalarTower.of_algebraMap_eq'
State after:
case h
R : Type u_1
S : Type u_2
inst✝¹⁵ : CommRing R
inst✝¹⁴ : CommRing S
inst✝¹³ : Algebra R S
p : Ideal R
inst✝¹² : p.IsMaximal
Rₚ : Type u_3
Sₚ : Type u_4
inst✝¹¹ : CommRing Rₚ
inst✝¹⁰ : CommRing Sₚ
inst✝⁹ : Algebra R Rₚ
inst✝⁸ : IsLocalization.AtPrime Rₚ p
inst✝⁷ : LocalRing Rₚ
inst✝⁶ : CommRing Sₚ
inst✝⁵ : Algebra S Sₚ
inst✝⁴ : Algebra R Sₚ
inst✝³ : Algebra Rₚ Sₚ
inst✝² : IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ
inst✝¹ : IsScalarTower R S Sₚ
inst✝ : IsScalarTower R Rₚ Sₚ
x : S
⊢ algebraMap R (Sₚ ⧸ pSₚ) = (algebraMap (Rₚ ⧸ maximalIdeal Rₚ) (Sₚ ⧸ pSₚ)).comp (algebraMap R (Rₚ ⧸ maximalIdeal Rₚ))

State before:
case h
R : Type u_1
S : Type u_2
inst✝¹⁵ : CommRing R
inst✝¹⁴ : CommRing S
inst✝¹³ : Algebra R S
p : Ideal R
inst✝¹² : p.IsMaximal
Rₚ : Type u_3
Sₚ : Type u_4
inst✝¹¹ : CommRing Rₚ
inst✝¹⁰ : CommRing Sₚ
inst✝⁹ : Algebra R Rₚ
inst✝⁸ : IsLocalization.AtPrime Rₚ p
inst✝⁷ : LocalRing Rₚ
inst✝⁶ : CommRing Sₚ
inst✝⁵ : Algebra S Sₚ
inst✝⁴ : Algebra R Sₚ
inst✝³ : Algebra Rₚ Sₚ
inst✝² : IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ
inst✝¹ : IsScalarTower R S Sₚ
inst✝ : IsScalarTower R Rₚ Sₚ
x : S
⊢ algebraMap R (Sₚ ⧸ pSₚ) = (algebraMap (Rₚ ⧸ maximalIdeal Rₚ) (Sₚ ⧸ pSₚ)).comp (algebraMap R (Rₚ ⧸ maximalIdeal Rₚ))
Tactic:
rw [IsScalarTower.algebraMap_eq R Rₚ (Rₚ ⧸ _), IsScalarTower.algebraMap_eq R Rₚ (Sₚ ⧸ _),
  ← RingHom.comp_assoc, ← IsScalarTower.algebraMap_eq Rₚ]
State after:
no goals

State before:
R : Type u_1
S : Type u_2
inst✝¹⁵ : CommRing R
inst✝¹⁴ : CommRing S
inst✝¹³ : Algebra R S
p : Ideal R
inst✝¹² : p.IsMaximal
Rₚ : Type u_3
Sₚ : Type u_4
inst✝¹¹ : CommRing Rₚ
inst✝¹⁰ : CommRing Sₚ
inst✝⁹ : Algebra R Rₚ
inst✝⁸ : IsLocalization.AtPrime Rₚ p
inst✝⁷ : LocalRing Rₚ
inst✝⁶ : CommRing Sₚ
inst✝⁵ : Algebra S Sₚ
inst✝⁴ : Algebra R Sₚ
inst✝³ : Algebra Rₚ Sₚ
inst✝² : IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ
inst✝¹ : IsScalarTower R S Sₚ
inst✝ : IsScalarTower R Rₚ Sₚ
x : S
this : IsScalarTower R (Rₚ ⧸ maximalIdeal Rₚ) (Sₚ ⧸ pSₚ)
⊢ (equivQuotMaximalIdealOfIsLocalization p Rₚ).symm
      ((Algebra.trace (Rₚ ⧸ maximalIdeal Rₚ) (Sₚ ⧸ pSₚ))
        ((quotMapEquivQuotMapMaximalIdealOfIsLocalization S p Rₚ Sₚ) ((Ideal.Quotient.mk pS) x))) =
    (equivQuotMaximalIdealOfIsLocalization p Rₚ).symm
      ((Algebra.trace (Rₚ ⧸ maximalIdeal Rₚ) (Sₚ ⧸ pSₚ)) ((algebraMap S (Sₚ ⧸ pSₚ)) x))
Tactic:
congr
State after:
no goals

State before:
case he
R : Type u_1
S : Type u_2
inst✝¹⁵ : CommRing R
inst✝¹⁴ : CommRing S
inst✝¹³ : Algebra R S
p : Ideal R
inst✝¹² : p.IsMaximal
Rₚ : Type u_3
Sₚ : Type u_4
inst✝¹¹ : CommRing Rₚ
inst✝¹⁰ : CommRing Sₚ
inst✝⁹ : Algebra R Rₚ
inst✝⁸ : IsLocalization.AtPrime Rₚ p
inst✝⁷ : LocalRing Rₚ
inst✝⁶ : CommRing Sₚ
inst✝⁵ : Algebra S Sₚ
inst✝⁴ : Algebra R Sₚ
inst✝³ : Algebra Rₚ Sₚ
inst✝² : IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ
inst✝¹ : IsScalarTower R S Sₚ
inst✝ : IsScalarTower R Rₚ Sₚ
x : S
this : IsScalarTower R (Rₚ ⧸ maximalIdeal Rₚ) (Sₚ ⧸ pSₚ)
⊢ (algebraMap (Rₚ ⧸ maximalIdeal Rₚ) (Sₚ ⧸ pSₚ)).comp ↑(equivQuotMaximalIdealOfIsLocalization p Rₚ) =
    (↑(quotMapEquivQuotMapMaximalIdealOfIsLocalization S p Rₚ Sₚ)).comp (algebraMap (R ⧸ p) (S ⧸ pS))
Tactic:
ext x
State after:
case he.h.a
R : Type u_1
S : Type u_2
inst✝¹⁵ : CommRing R
inst✝¹⁴ : CommRing S
inst✝¹³ : Algebra R S
p : Ideal R
inst✝¹² : p.IsMaximal
Rₚ : Type u_3
Sₚ : Type u_4
inst✝¹¹ : CommRing Rₚ
inst✝¹⁰ : CommRing Sₚ
inst✝⁹ : Algebra R Rₚ
inst✝⁸ : IsLocalization.AtPrime Rₚ p
inst✝⁷ : LocalRing Rₚ
inst✝⁶ : CommRing Sₚ
inst✝⁵ : Algebra S Sₚ
inst✝⁴ : Algebra R Sₚ
inst✝³ : Algebra Rₚ Sₚ
inst✝² : IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ
inst✝¹ : IsScalarTower R S Sₚ
inst✝ : IsScalarTower R Rₚ Sₚ
x✝ : S
this : IsScalarTower R (Rₚ ⧸ maximalIdeal Rₚ) (Sₚ ⧸ pSₚ)
x : R
⊢ (((algebraMap (Rₚ ⧸ maximalIdeal Rₚ) (Sₚ ⧸ pSₚ)).comp ↑(equivQuotMaximalIdealOfIsLocalization p Rₚ)).comp
        (Ideal.Quotient.mk p))
      x =
    (((↑(quotMapEquivQuotMapMaximalIdealOfIsLocalization S p Rₚ Sₚ)).comp (algebraMap (R ⧸ p) (S ⧸ pS))).comp
        (Ideal.Quotient.mk p))
      x

State before:
case he.h.a
R : Type u_1
S : Type u_2
inst✝¹⁵ : CommRing R
inst✝¹⁴ : CommRing S
inst✝¹³ : Algebra R S
p : Ideal R
inst✝¹² : p.IsMaximal
Rₚ : Type u_3
Sₚ : Type u_4
inst✝¹¹ : CommRing Rₚ
inst✝¹⁰ : CommRing Sₚ
inst✝⁹ : Algebra R Rₚ
inst✝⁸ : IsLocalization.AtPrime Rₚ p
inst✝⁷ : LocalRing Rₚ
inst✝⁶ : CommRing Sₚ
inst✝⁵ : Algebra S Sₚ
inst✝⁴ : Algebra R Sₚ
inst✝³ : Algebra Rₚ Sₚ
inst✝² : IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ
inst✝¹ : IsScalarTower R S Sₚ
inst✝ : IsScalarTower R Rₚ Sₚ
x✝ : S
this : IsScalarTower R (Rₚ ⧸ maximalIdeal Rₚ) (Sₚ ⧸ pSₚ)
x : R
⊢ (((algebraMap (Rₚ ⧸ maximalIdeal Rₚ) (Sₚ ⧸ pSₚ)).comp ↑(equivQuotMaximalIdealOfIsLocalization p Rₚ)).comp
        (Ideal.Quotient.mk p))
      x =
    (((↑(quotMapEquivQuotMapMaximalIdealOfIsLocalization S p Rₚ Sₚ)).comp (algebraMap (R ⧸ p) (S ⧸ pS))).comp
        (Ideal.Quotient.mk p))
      x
Tactic:
simp only [equivQuotMaximalIdealOfIsLocalization, RingHom.quotientKerEquivOfSurjective,
  RingEquiv.coe_ringHom_trans, RingHom.coe_comp, RingHom.coe_coe, Function.comp_apply,
  Ideal.quotEquivOfEq_mk, RingHom.quotientKerEquivOfRightInverse.apply, RingHom.kerLift_mk,
  quotMapEquivQuotMapMaximalIdealOfIsLocalization,
  Ideal.Quotient.algebraMap_quotient_map_quotient]
State after:
case he.h.a
R : Type u_1
S : Type u_2
inst✝¹⁵ : CommRing R
inst✝¹⁴ : CommRing S
inst✝¹³ : Algebra R S
p : Ideal R
inst✝¹² : p.IsMaximal
Rₚ : Type u_3
Sₚ : Type u_4
inst✝¹¹ : CommRing Rₚ
inst✝¹⁰ : CommRing Sₚ
inst✝⁹ : Algebra R Rₚ
inst✝⁸ : IsLocalization.AtPrime Rₚ p
inst✝⁷ : LocalRing Rₚ
inst✝⁶ : CommRing Sₚ
inst✝⁵ : Algebra S Sₚ
inst✝⁴ : Algebra R Sₚ
inst✝³ : Algebra Rₚ Sₚ
inst✝² : IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ
inst✝¹ : IsScalarTower R S Sₚ
inst✝ : IsScalarTower R Rₚ Sₚ
x✝ : S
this : IsScalarTower R (Rₚ ⧸ maximalIdeal Rₚ) (Sₚ ⧸ pSₚ)
x : R
⊢ (algebraMap (Rₚ ⧸ maximalIdeal Rₚ) (Sₚ ⧸ pSₚ)) ((algebraMap R (Rₚ ⧸ maximalIdeal Rₚ)) x) =
    (algebraMap S (Sₚ ⧸ pSₚ)) ((algebraMap R S) x)

State before:
case he.h.a
R : Type u_1
S : Type u_2
inst✝¹⁵ : CommRing R
inst✝¹⁴ : CommRing S
inst✝¹³ : Algebra R S
p : Ideal R
inst✝¹² : p.IsMaximal
Rₚ : Type u_3
Sₚ : Type u_4
inst✝¹¹ : CommRing Rₚ
inst✝¹⁰ : CommRing Sₚ
inst✝⁹ : Algebra R Rₚ
inst✝⁸ : IsLocalization.AtPrime Rₚ p
inst✝⁷ : LocalRing Rₚ
inst✝⁶ : CommRing Sₚ
inst✝⁵ : Algebra S Sₚ
inst✝⁴ : Algebra R Sₚ
inst✝³ : Algebra Rₚ Sₚ
inst✝² : IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ
inst✝¹ : IsScalarTower R S Sₚ
inst✝ : IsScalarTower R Rₚ Sₚ
x✝ : S
this : IsScalarTower R (Rₚ ⧸ maximalIdeal Rₚ) (Sₚ ⧸ pSₚ)
x : R
⊢ (algebraMap (Rₚ ⧸ maximalIdeal Rₚ) (Sₚ ⧸ pSₚ)) ((algebraMap R (Rₚ ⧸ maximalIdeal Rₚ)) x) =
    (algebraMap S (Sₚ ⧸ pSₚ)) ((algebraMap R S) x)
Tactic:
rw [← IsScalarTower.algebraMap_apply, ← IsScalarTower.algebraMap_apply]
State after:
no goals