mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.Smooth.IntegralClosure

Goal state before:
pp: R : Type u_1
S : Type u_2
B : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
inst✝² : CommRing B
inst✝¹ : Algebra R B
inst✝ : Algebra.IsStandardEtale R S
𝓟 : StandardEtalePresentation R S
n : ℕ
𝓟' : StandardEtalePresentation B (B ⊗[R] S) := 𝓟.baseChange
e✝ : B ⊗[R] S ≃ₐ[B] Localization.Away ((AdjoinRoot.mk 𝓟'.f) 𝓟'.g) := 𝓟'.equivRing.trans 𝓟'.equivAwayAdjoinRoot
a : AdjoinRoot 𝓟'.f
m : ℕ
hx✝ :
  IsIntegral S
    ((Algebra.TensorProduct.comm R B S)
      (e✝.symm
        (IsLocalization.mk' (Localization.Away ((AdjoinRoot.mk 𝓟'.f) 𝓟'.g)) a
          ⟨(fun x ↦ (AdjoinRoot.mk 𝓟'.f) 𝓟'.g ^ x) m, ⋯⟩)))
hx :
  ∀ (m_1 : ℕ),
    IsIntegral R
      ((aeval 𝓟.x) 𝓟.g ^ (n + m_1) •
        (Algebra.TensorProduct.comm R B S)
          (e✝.symm
            (IsLocalization.mk' (Localization.Away ((AdjoinRoot.mk 𝓟'.f) 𝓟'.g)) a
              ⟨(fun x ↦ (AdjoinRoot.mk 𝓟'.f) 𝓟'.g ^ x) m, ⋯⟩)))
this : Module.Finite R (AdjoinRoot 𝓟.f)
e : AdjoinRoot 𝓟.f →ₐ[R] AdjoinRoot 𝓟'.f := AdjoinRoot.mapAlgHom (Algebra.ofId R B) 𝓟.f 𝓟'.f ⋯
⊢ (AdjoinRoot.mk 𝓟'.f).comp (mapRingHom (algebraMap R B)) = e.comp (AdjoinRoot.mk 𝓟.f)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.ZariskisMainTheorem

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
φ : R[X] →ₐ[R] S
t : S
p : R[X]
ht : φ.IsIntegralElem t
hp : φ p * t ∈ φ.range
a : R := p.leadingCoeff
R' : Type u_1 := Localization.Away a
S' : Type u_2 := Localization.Away ((algebraMap R S) a)
this✝ : Algebra R' S' := (Localization.awayMap (algebraMap R S) a).toAlgebra
this : IsScalarTower R R' S'
ha : IsUnit ((algebraMap R R') a)
H : (aeval ((algebraMap S S') (φ X))).comp (mapRingHom (algebraMap R R')) = (algebraMap S S').comp ↑φ
q : R'[X]
hq : IsIntegral R' ((algebraMap S S') t - (aeval ((algebraMap S S') (φ X))) q)
n : ℕ
q' : R[X]
e : map (algebraMap R R') q' = (fun x ↦ a ^ x) n • q
⊢ IsIntegral R' ((algebraMap S S') (a ^ n • t - φ q'))

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Smooth.IntegralClosure

Goal state before:
pp: R : Type u_1
S : Type u_2
B : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
inst✝² : CommRing B
inst✝¹ : Algebra R B
inst✝ : Algebra.IsStandardEtale R S
𝓟 : StandardEtalePresentation R S
n : ℕ
𝓟' : StandardEtalePresentation B (B ⊗[R] S) := 𝓟.baseChange
e : B ⊗[R] S ≃ₐ[B] Localization.Away ((AdjoinRoot.mk 𝓟'.f) 𝓟'.g) := 𝓟'.equivRing.trans 𝓟'.equivAwayAdjoinRoot
a : AdjoinRoot 𝓟'.f
m : ℕ
hx✝ :
  IsIntegral S
    ((Algebra.TensorProduct.comm R B S)
      (e.symm
        (IsLocalization.mk' (Localization.Away ((AdjoinRoot.mk 𝓟'.f) 𝓟'.g)) a
          ⟨(fun x ↦ (AdjoinRoot.mk 𝓟'.f) 𝓟'.g ^ x) m, ⋯⟩)))
hx :
  ∀ (m_1 : ℕ),
    IsIntegral R
      ((aeval 𝓟.x) 𝓟.g ^ (n + m_1) •
        (Algebra.TensorProduct.comm R B S)
          (e.symm
            (IsLocalization.mk' (Localization.Away ((AdjoinRoot.mk 𝓟'.f) 𝓟'.g)) a
              ⟨(fun x ↦ (AdjoinRoot.mk 𝓟'.f) 𝓟'.g ^ x) m, ⋯⟩)))
⊢ IsIntegral R ((AdjoinRoot.mk 𝓟'.f) 𝓟'.g)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.ZariskisMainTheorem

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
φ : R[X] →ₐ[R] S
t : S
p : R[X]
ht : φ.IsIntegralElem t
hp : φ p * t ∈ φ.range
⊢ ∃ q n, IsIntegral R (p.leadingCoeff ^ n • t - φ q)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Unramified.LocalStructure

Goal state before:
pp: R : Type u_1
S : Type u_3
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
Q : Ideal S
inst✝⁴ : Q.IsPrime
inst✝³ : Module.Finite R S
inst✝² : IsUnramifiedAt R Q
inst✝¹ : Algebra (Localization.AtPrime (Ideal.under R Q)) (Localization.AtPrime Q)
inst✝ : Localization.AtPrime.IsLiesOverAlgebra (Ideal.under R Q) Q
p : Ideal R := Ideal.under R Q
this✝⁴ : p.IsPrime
this✝³ : Module p.ResidueField (p.Fiber S) := leftModule
this✝² : SMul p.ResidueField (p.Fiber S) := this✝³.toSMul
this✝¹ : IsScalarTower p.ResidueField (p.Fiber S) (p.Fiber S) := IsScalarTower.right
this✝ : Module.Finite p.ResidueField (p.Fiber S) := Module.Finite.base_change R p.ResidueField S
this : IsArtinianRing (p.Fiber S)
α : ↑(p.primesOver S) ≃o PrimeSpectrum (p.Fiber S) := PrimeSpectrum.primesOverOrderIsoFiber R S p
x : p.ResidueField ⊗[R] S
φ : p.Fiber S →ₐ[p.ResidueField] Q.ResidueField :=
  TensorProduct.lift (ofId p.ResidueField Q.ResidueField) (IsScalarTower.toAlgHom R S Q.ResidueField) ⋯
hx0 : φ x ≠ 0
hx : p.ResidueField[φ x] = ⊤
r : p.Fiber S
hrQ : r ∉ (α ⟨Q, ⋯⟩).asIdeal
hrid : IsIdempotentElem r
hr : ∀ (q : Ideal (p.Fiber S)), q.IsPrime → q ≠ (α ⟨Q, ⋯⟩).asIdeal → r ∈ q
s : R
hsQ : s ∉ p
t : S
e : s • (r * x) = 1 ⊗ₜ[R] t
hsQ' : (algebraMap R Q.ResidueField) s ≠ 0
hrQ' : φ r = 1
e' : (algebraMap R Q.ResidueField) s * φ x = (algebraMap S Q.ResidueField) t
⊢ φ x = ((algebraMap R p.ResidueField) s)⁻¹ • (algebraMap S Q.ResidueField) t

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.ZariskisMainTheorem

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
φ : R[X] →ₐ[R] S
t : S
p : R[X]
hRS : integralClosure R S = ⊥
hφ : φ.Finite
hp : φ p * t ∈ conductor R (φ X)
algInst✝ : Algebra R[X] S := φ.toAlgebra
algebraizeInst✝ : Module.Finite R[X] S
this : IsScalarTower R R[X] S
x : S
⊢ φ p * t * x ∈ φ.range

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.ZariskisMainTheorem

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
φ : R[X] →ₐ[R] S
t : S
p : R[X]
hRS : integralClosure R S = ⊥
hφ : φ.Finite
hp : φ p * t ∈ conductor R (φ X)
algInst✝ : Algebra R[X] S := φ.toAlgebra
algebraizeInst✝ : Module.Finite R[X] S
this✝ : IsScalarTower R R[X] S
x : S
this : φ p * t * x ∈ φ.range
⊢ φ p * (t * x) ∈ φ.range

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.ZariskisMainTheorem

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
φ : R[X] →ₐ[R] S
t : S
p : R[X]
hRS : integralClosure R S = ⊥
hφ : φ.Finite
hp : φ p * t ∈ conductor R (φ X)
algInst✝ : Algebra R[X] S := φ.toAlgebra
algebraizeInst✝ : Module.Finite R[X] S
this✝ : IsScalarTower R R[X] S
x : S
this : φ p * t * x ∈ φ.range
q : R[X]
n : ℕ
hn : IsIntegral R (p.leadingCoeff ^ n • (t * x) - φ q)
r : R
hr : (algebraMap R S) r = p.leadingCoeff ^ n • (t * x) - φ q
⊢ φ.toRingHom (C r + q) = p.leadingCoeff ^ n • (t * x)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.ZariskisMainTheorem

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
φ : R[X] →ₐ[R] S
t : S
p : R[X]
hRS : integralClosure R S = ⊥
hφ : φ.Finite
hp : φ p * t ∈ conductor R (φ X)
algInst✝ : Algebra R[X] S := φ.toAlgebra
algebraizeInst✝ : Module.Finite R[X] S
this : IsScalarTower R R[X] S
x : S
⊢ ∃ n, p.leadingCoeff ^ n • (t * x) ∈ φ.range

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Unramified.LocalStructure

Goal state before:
pp: R : Type u_1
S : Type u_3
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
Q : Ideal S
inst✝⁴ : Q.IsPrime
inst✝³ : Module.Finite R S
inst✝² : IsUnramifiedAt R Q
inst✝¹ : Algebra (Localization.AtPrime (Ideal.under R Q)) (Localization.AtPrime Q)
inst✝ : Localization.AtPrime.IsLiesOverAlgebra (Ideal.under R Q) Q
⊢ ∃ t ∉ Q,
    (∀ Q' ∈ (Ideal.under R Q).primesOver S, Q' ≠ Q → t ∈ Q') ∧
      (Ideal.under R Q).ResidueField[(algebraMap S Q.ResidueField) t] = ⊤

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Smooth.IntegralClosure

Goal state before:
pp: R : Type u_1
S : Type u_2
B : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
inst✝² : CommRing B
inst✝¹ : Algebra R B
inst✝ : Algebra.IsStandardEtale R S
𝓟 : StandardEtalePresentation R S
n : ℕ
𝓟' : StandardEtalePresentation B (B ⊗[R] S) := 𝓟.baseChange
e : B ⊗[R] S ≃ₐ[B] Localization.Away ((AdjoinRoot.mk 𝓟'.f) 𝓟'.g) := 𝓟'.equivRing.trans 𝓟'.equivAwayAdjoinRoot
a : AdjoinRoot 𝓟'.f
m : ℕ
hx✝ :
  IsIntegral S
    ((Algebra.TensorProduct.comm R B S)
      (e.symm
        (IsLocalization.mk' (Localization.Away ((AdjoinRoot.mk 𝓟'.f) 𝓟'.g)) a
          ⟨(fun x ↦ (AdjoinRoot.mk 𝓟'.f) 𝓟'.g ^ x) m, ⋯⟩)))
hx :
  ∀ (m_1 : ℕ),
    IsIntegral R
      ((aeval 𝓟.x) 𝓟.g ^ (n + m_1) •
        (Algebra.TensorProduct.comm R B S)
          (e.symm
            (IsLocalization.mk' (Localization.Away ((AdjoinRoot.mk 𝓟'.f) 𝓟'.g)) a
              ⟨(fun x ↦ (AdjoinRoot.mk 𝓟'.f) 𝓟'.g ^ x) m, ⋯⟩)))
hfg : IsIntegral R ((AdjoinRoot.mk 𝓟'.f) 𝓟'.g)
g : R[X]
⊢ e (1 ⊗ₜ[R] (aeval 𝓟.x) g) =
    (algebraMap (AdjoinRoot 𝓟'.f) (Localization.Away ((AdjoinRoot.mk 𝓟'.f) 𝓟'.g)))
      ((AdjoinRoot.mk 𝓟'.f) (Polynomial.map (algebraMap R B) g))

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.ZariskisMainTheorem

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
φ : R[X] →ₐ[R] S
t : S
p : R[X]
hRS : integralClosure R S = ⊥
hφ : φ.Finite
hp : φ p * t ∈ conductor R (φ X)
⊢ ∃ n, p.leadingCoeff ^ n • t ∈ conductor R (φ X)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Unramified.LocalStructure

Goal state before:
pp: R : Type u_1
S : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
Q : Ideal S
inst✝² : Q.IsPrime
inst✝¹ : Module.Finite R S
inst✝ : IsUnramifiedAt R Q
this✝¹ : Algebra (Localization.AtPrime (Ideal.under R Q)) (Localization.AtPrime Q) :=
  Localization.AtPrime.algebraOfLiesOver (Ideal.under R Q) Q
t : S
htQ : t ∉ Q
htQ' : ∀ Q' ∈ (Ideal.under R Q).primesOver S, Q' ≠ Q → t ∈ Q'
ht : (Ideal.under R Q).ResidueField[(algebraMap S Q.ResidueField) t] = ⊤
p : Ideal R := Ideal.under R Q
this✝ : Algebra (Localization.AtPrime p) (Localization.AtPrime (Ideal.under (↥R[t]) Q)) :=
  Localization.AtPrime.algebraOfLiesOver p (Ideal.under (↥R[t]) Q)
this : Algebra (Localization.AtPrime (Ideal.under (↥R[t]) Q)) (Localization.AtPrime Q) :=
  Localization.AtPrime.algebraOfLiesOver (Ideal.under (↥R[t]) Q) Q
Q' : Ideal S
e : Q' ∈ {Q}
⊢ Q' ∈ (Ideal.under (↥R[t]) Q).primesOver S

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.ZariskisMainTheorem

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
φ : R[X] →ₐ[R] S
t : S
hRS : integralClosure R S = ⊥
hφ : φ.Finite
i : ℕ
this : ∀ (p : R[X]), φ p * t ∈ (conductor R (φ X)).radical → p.leadingCoeff • t ∈ (conductor R (φ X)).radical
p : R[X]
hp : φ p * t ∈ (conductor R (φ X)).radical
n : ℕ
IH :
  ∀ m < n + 1,
    ∀ (p : R[X]), φ p * t ∈ (conductor R (φ X)).radical → p.natDegree = m → p.coeff i • t ∈ (conductor R (φ X)).radical
hpn : p.natDegree = n + 1
hi' : i ≠ p.natDegree
⊢ φ p.eraseLead * t ∈ (conductor R (φ X)).radical

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.ZariskisMainTheorem

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
φ : R[X] →ₐ[R] S
t : S
hRS : integralClosure R S = ⊥
hφ : φ.Finite
i : ℕ
this✝ : ∀ (p : R[X]), φ p * t ∈ (conductor R (φ X)).radical → p.leadingCoeff • t ∈ (conductor R (φ X)).radical
p : R[X]
hp : φ p * t ∈ (conductor R (φ X)).radical
n : ℕ
IH :
  ∀ m < n + 1,
    ∀ (p : R[X]), φ p * t ∈ (conductor R (φ X)).radical → p.natDegree = m → p.coeff i • t ∈ (conductor R (φ X)).radical
hpn : p.natDegree = n + 1
hi' : i ≠ p.natDegree
this : φ p.eraseLead * t ∈ (conductor R (φ X)).radical
⊢ p.natDegree - 1 < n + 1

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.ZariskisMainTheorem

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
φ : R[X] →ₐ[R] S
t : S
p✝ : R[X]
hRS : integralClosure R S = ⊥
hφ : φ.Finite
p : R[X]
i : ℕ
hi : i = p.natDegree
n : ℕ
hn : (φ p * t) ^ n ∈ conductor R (φ X)
⊢ φ (p ^ n) * t ^ n ∈ conductor R (φ X)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Unramified.LocalStructure

Goal state before:
pp: R : Type u_1
S : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
Q : Ideal S
inst✝² : Q.IsPrime
inst✝¹ : Module.Finite R S
inst✝ : IsUnramifiedAt R Q
⊢ ∃ x,
    (Ideal.under (↥R[x]) Q).primesOver S = {Q} ∧
      Function.Bijective ⇑(algebraMap (Ideal.under (↥R[x]) Q).ResidueField Q.ResidueField)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Smooth.IntegralClosure

Goal state before:
pp: R : Type u_1
S : Type u_2
B : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
inst✝² : CommRing B
inst✝¹ : Algebra R B
inst✝ : Algebra.IsStandardEtale R S
𝓟 : StandardEtalePresentation R S
n : ℕ
𝓟' : StandardEtalePresentation B (B ⊗[R] S) := 𝓟.baseChange
e : B ⊗[R] S ≃ₐ[B] Localization.Away ((AdjoinRoot.mk 𝓟'.f) 𝓟'.g) := 𝓟'.equivRing.trans 𝓟'.equivAwayAdjoinRoot
a : AdjoinRoot 𝓟'.f
m : ℕ
hx✝ :
  IsIntegral S
    ((Algebra.TensorProduct.comm R B S)
      (e.symm
        (IsLocalization.mk' (Localization.Away ((AdjoinRoot.mk 𝓟'.f) 𝓟'.g)) a
          ⟨(fun x ↦ (AdjoinRoot.mk 𝓟'.f) 𝓟'.g ^ x) m, ⋯⟩)))
hx :
  ∀ (m_1 : ℕ),
    IsIntegral R
      ((aeval 𝓟.x) 𝓟.g ^ (n + m_1) •
        (Algebra.TensorProduct.comm R B S)
          (e.symm
            (IsLocalization.mk' (Localization.Away ((AdjoinRoot.mk 𝓟'.f) 𝓟'.g)) a
              ⟨(fun x ↦ (AdjoinRoot.mk 𝓟'.f) 𝓟'.g ^ x) m, ⋯⟩)))
hfg : IsIntegral R ((AdjoinRoot.mk 𝓟'.f) 𝓟'.g)
heg :
  ∀ (g : R[X]),
    e (1 ⊗ₜ[R] (aeval 𝓟.x) g) =
      (algebraMap (AdjoinRoot 𝓟'.f) (Localization.Away ((AdjoinRoot.mk 𝓟'.f) 𝓟'.g)))
        ((AdjoinRoot.mk 𝓟'.f) (Polynomial.map (algebraMap R B) g))
⊢ ∀ (k : ℕ),
    e (1 ⊗ₜ[R] ((aeval 𝓟.x) 𝓟.g ^ k)) =
      (algebraMap (AdjoinRoot 𝓟'.f) (Localization.Away ((AdjoinRoot.mk 𝓟'.f) 𝓟'.g))) ((AdjoinRoot.mk 𝓟'.f) 𝓟'.g ^ k)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Unramified.LocalStructure

Goal state before:
pp: R : Type u_1
S : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
Q : Ideal S
inst✝² : Q.IsPrime
inst✝¹ : Module.Finite R S
inst✝ : IsUnramifiedAt R Q
x : S
hQ' : (Ideal.under (↥R[x]) Q).primesOver S = {Q}
hQ'Q : Function.Bijective ⇑(algebraMap (Ideal.under (↥R[x]) Q).ResidueField Q.ResidueField)
S' : Subalgebra R S := R[x]
Q' : Ideal ↥S' := Ideal.under (↥S') Q
this✝³ : Algebra (Localization.AtPrime Q') (Localization.AtPrime Q) := Localization.AtPrime.algebraOfLiesOver Q' Q
this✝² : Module.Finite (↥S') S
this✝¹ : IsUnramifiedAt (↥S') Q
hφ : Function.Bijective ⇑(Localization.localRingHom Q' Q ↑S'.val ⋯)
r : ↥R[x]
hrQ' : r ∉ Ideal.under (↥R[x]) Q
H : ∀ (r' : ↥R[x]), r ∣ r' → Function.Bijective ⇑(Localization.awayMap (algebraMap (↥R[x]) S) r')
this✝ : Module.Finite R ↥S'
this : IsUnramifiedAt R Q'
⊢ Subalgebra.map S'.val R[⟨x, ⋯⟩] = Subalgebra.map S'.val ⊤

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.ZariskisMainTheorem

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
φ : R[X] →ₐ[R] S
t : S
p : R[X]
hRS : integralClosure R S = ⊥
hφ : φ.Finite
hp : φ p * t ∈ (conductor R (φ X)).radical
i : ℕ
⊢ p.coeff i • t ∈ (conductor R (φ X)).radical

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Unramified.LocalStructure

Goal state before:
pp: R : Type u_1
S : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
Q : Ideal S
inst✝² : Q.IsPrime
inst✝¹ : Module.Finite R S
inst✝ : IsUnramifiedAt R Q
x : S
hQ' : (Ideal.under (↥R[x]) Q).primesOver S = {Q}
hQ'Q : Function.Bijective ⇑(algebraMap (Ideal.under (↥R[x]) Q).ResidueField Q.ResidueField)
S' : Subalgebra R S := R[x]
Q' : Ideal ↥S' := Ideal.under (↥S') Q
this✝³ : Algebra (Localization.AtPrime Q') (Localization.AtPrime Q) := Localization.AtPrime.algebraOfLiesOver Q' Q
this✝² : Module.Finite (↥S') S
this✝¹ : IsUnramifiedAt (↥S') Q
hφ : Function.Bijective ⇑(Localization.localRingHom Q' Q ↑S'.val ⋯)
r : ↥R[x]
hrQ' : r ∉ Ideal.under (↥R[x]) Q
H : ∀ (r' : ↥R[x]), r ∣ r' → Function.Bijective ⇑(Localization.awayMap (algebraMap (↥R[x]) S) r')
this✝ : Module.Finite R ↥S'
this : IsUnramifiedAt R Q'
f : ↥S'
hfQ' : f ∉ Q'
hf : HasStandardEtaleSurjectionOn R f
⊢ f ∣ f * r

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.ZariskisMainTheorem

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
hRS : integralClosure R S = ⊥
x : S
hx : (aeval x).Finite
u : S
p : R[X]
e : (aeval x) p * u ∈ (conductor R x).radical
i : ℕ
⊢ (aeval x) p * u ∈ (conductor R ((aeval x) X)).radical

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Unramified.LocalStructure

Goal state before:
pp: R : Type u_1
S : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
Q : Ideal S
inst✝² : Q.IsPrime
inst✝¹ : Module.Finite R S
inst✝ : IsUnramifiedAt R Q
x : S
hQ' : (Ideal.under (↥R[x]) Q).primesOver S = {Q}
hQ'Q : Function.Bijective ⇑(algebraMap (Ideal.under (↥R[x]) Q).ResidueField Q.ResidueField)
S' : Subalgebra R S := R[x]
Q' : Ideal ↥S' := Ideal.under (↥S') Q
this✝³ : Algebra (Localization.AtPrime Q') (Localization.AtPrime Q) := Localization.AtPrime.algebraOfLiesOver Q' Q
this✝² : Module.Finite (↥S') S
this✝¹ : IsUnramifiedAt (↥S') Q
hφ : Function.Bijective ⇑(Localization.localRingHom Q' Q ↑S'.val ⋯)
r : ↥R[x]
hrQ' : r ∉ Ideal.under (↥R[x]) Q
H : ∀ (r' : ↥R[x]), r ∣ r' → Function.Bijective ⇑(Localization.awayMap (algebraMap (↥R[x]) S) r')
this✝ : Module.Finite R ↥S'
this : IsUnramifiedAt R Q'
f : ↥S'
hfQ' : f ∉ Q'
hf : HasStandardEtaleSurjectionOn R f
P : StandardEtalePair R
φ : P.Ring →ₐ[R] Localization.Away (f * r)
hP : Function.Surjective ⇑φ
⊢ r ∣ f * r

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.ZariskisMainTheorem

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
hRS : integralClosure R S = ⊥
x : S
hx : (aeval x).Finite
⊢ IsStronglyTranscendental R ((Ideal.Quotient.mk (conductor R x).radical) x)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.ZariskisMainTheorem

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : FaithfulSMul R S
inst✝³ : IsIntegrallyClosed R
inst✝² : IsDomain S
x : S
P : Ideal S
inst✝¹ : P.IsPrime
inst✝ : WeaklyQuasiFiniteAt R P
hx : IsStronglyTranscendental R x
this : IsDomain R
f : R[X] →ₐ[R] S
hf : aeval x = f
hx' : f.Finite
hf' : Function.Injective ⇑f
⊢ f X = x

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Unramified.LocalStructure

Goal state before:
pp: R : Type u_1
S : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
Q : Ideal S
inst✝² : Q.IsPrime
inst✝¹ : Module.Finite R S
inst✝ : IsUnramifiedAt R Q
x : S
hQ' : (Ideal.under (↥R[x]) Q).primesOver S = {Q}
hQ'Q : Function.Bijective ⇑(algebraMap (Ideal.under (↥R[x]) Q).ResidueField Q.ResidueField)
S' : Subalgebra R S := R[x]
Q' : Ideal ↥S' := Ideal.under (↥S') Q
this✝³ : Algebra (Localization.AtPrime Q') (Localization.AtPrime Q) := Localization.AtPrime.algebraOfLiesOver Q' Q
this✝² : Module.Finite (↥S') S
this✝¹ : IsUnramifiedAt (↥S') Q
hφ : Function.Bijective ⇑(Localization.localRingHom Q' Q ↑S'.val ⋯)
r : ↥R[x]
hrQ' : r ∉ Ideal.under (↥R[x]) Q
H : ∀ (r' : ↥R[x]), r ∣ r' → Function.Bijective ⇑(Localization.awayMap (algebraMap (↥R[x]) S) r')
this✝ : Module.Finite R ↥S'
this : IsUnramifiedAt R Q'
f : ↥S'
hfQ' : f ∉ Q'
hf : HasStandardEtaleSurjectionOn R f
P : StandardEtalePair R
φ : P.Ring →ₐ[R] Localization.Away (f * r)
hP : Function.Surjective ⇑φ
⊢ Function.Surjective ⇑((Localization.awayMapₐ (IsScalarTower.toAlgHom R (↥S') S) (f * r)).comp φ)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Unramified.LocalStructure

Goal state before:
pp: R : Type u_1
S : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
Q : Ideal S
inst✝² : Q.IsPrime
inst✝¹ : Module.Finite R S
inst✝ : IsUnramifiedAt R Q
⊢ ∃ f ∉ Q, HasStandardEtaleSurjectionOn R f

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Unramified.LocalStructure

Goal state before:
pp: R : Type u_1
S : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
Q : Ideal S
inst✝² : Q.IsPrime
inst✝¹ : FiniteType R S
inst✝ : IsUnramifiedAt R Q
this :
  ∀ {R : Type u_1} {S : Type u_3} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] (Q : Ideal S)
    [inst_3 : Q.IsPrime] [FiniteType R S] [IsUnramifiedAt R Q],
    Unramified R S → ∃ f ∉ Q, HasStandardEtaleSurjectionOn R f
H : ¬Unramified R S
s : S
hsQ : s ∉ Q
hs : FormallyUnramified R (Localization.Away s)
⊢ Disjoint ↑(Submonoid.powers s) ↑Q

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.ZariskisMainTheorem

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : FaithfulSMul R S
inst✝³ : IsIntegrallyClosed R
inst✝² : IsDomain S
P : Ideal S
inst✝¹ : P.IsPrime
inst✝ : WeaklyQuasiFiniteAt R P
this✝² : IsDomain R
f : R[X] →ₐ[R] S
hx' : f.Finite
hf' : Function.Injective ⇑f
hx : IsStronglyTranscendental R (f X)
hf : aeval (f X) = f
this✝¹ : Algebra R[X] S := f.toAlgebra
this✝ : IsScalarTower R R[X] S
this : Module.Finite R[X] S
p : R[X]
hp : (algebraMap R[X] S) p = 0
hp' : ¬p = 0
⊢ (aeval (f X)) p = 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.ZariskisMainTheorem

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : FaithfulSMul R S
inst✝³ : IsIntegrallyClosed R
inst✝² : IsDomain S
P : Ideal S
inst✝¹ : P.IsPrime
inst✝ : WeaklyQuasiFiniteAt R P
this✝² : IsDomain R
f : R[X] →ₐ[R] S
hx' : f.Finite
hf' : Function.Injective ⇑f
hx : IsStronglyTranscendental R (f X)
hf : aeval (f X) = f
this✝¹ : Algebra R[X] S := f.toAlgebra
this✝ : IsScalarTower R R[X] S
this : Module.Finite R[X] S
⊢ FaithfulSMul R[X] S

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.ZariskisMainTheorem

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : FaithfulSMul R S
inst✝³ : IsIntegrallyClosed R
inst✝² : IsDomain S
x : S
hx' : (aeval x).Finite
P : Ideal S
inst✝¹ : P.IsPrime
inst✝ : WeaklyQuasiFiniteAt R P
⊢ ¬IsStronglyTranscendental R x

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Unramified.LocalStructure

Goal state before:
pp: R : Type u_1
S : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
Q : Ideal S
inst✝² : Q.IsPrime
inst✝¹ : FiniteType R S
inst✝ : IsUnramifiedAt R Q
this :
  ∀ {R : Type u_1} {S : Type u_3} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] (Q : Ideal S)
    [inst_3 : Q.IsPrime] [FiniteType R S] [IsUnramifiedAt R Q],
    Unramified R S → ∃ f ∉ Q, HasStandardEtaleSurjectionOn R f
H✝ : ¬Unramified R S
s : S
hsQ : s ∉ Q
hs : FormallyUnramified R (Localization.Away s)
hQ : (Ideal.map (algebraMap S (Localization.Away s)) Q).IsPrime
inst : Unramified R (Localization.Away s)
f : S
t : ↥(Submonoid.powers s)
hf : IsLocalization.mk' (Localization.Away s) f t ∉ Ideal.map (algebraMap S (Localization.Away s)) Q
H : HasStandardEtaleSurjectionOn R (IsLocalization.mk' (Localization.Away s) f t)
⊢ (algebraMap S (Localization.Away s)) f =
    IsLocalization.mk' (Localization.Away s) f t * (algebraMap S (Localization.Away s)) ↑t

Tactic:
by

Goal state after: