mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.Extension.Cotangent.Free

Goal state before:
pp: case hsp
R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
ι : Type u_3
σ : Type u_4
P : PreSubmersivePresentation R S ι σ
inst✝ : Finite σ
b : Module.Basis σ S P.toExtension.Cotangent
hb : ∀ (r : σ), b r = Extension.Cotangent.mk ⟨P.relation r, ⋯⟩
h : Function.Bijective ⇑(P.cotangentRestrict ⋯)
heq :
  (fun j i ↦ (aeval P.val) ((pderiv (P.map i)) (P.relation j))) =
    ⇑(Finsupp.linearEquivFunOnFinite S S σ) ∘ ⇑(P.cotangentRestrict ⋯) ∘ ⇑b
⊢ Submodule.span S (Set.range fun j i ↦ (aeval P.val) ((pderiv (P.map i)) (P.relation j))) = ⊤

Tactic:
·

Goal state after:
case hsp
R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
ι : Type u_3
σ : Type u_4
P : PreSubmersivePresentation R S ι σ
inst✝ : Finite σ
b : Module.Basis σ S P.toExtension.Cotangent
hb : ∀ (r : σ), b r = Extension.Cotangent.mk ⟨P.relation r, ⋯⟩
h : Function.Bijective ⇑(P.cotangentRestrict ⋯)
heq :
  (fun j i ↦ (aeval P.val) ((pderiv (P.map i)) (P.relation j))) =
    ⇑(Finsupp.linearEquivFunOnFinite S S σ) ∘ ⇑(P.cotangentRestrict ⋯) ∘ ⇑b
⊢ Submodule.span S (Set.range fun j i ↦ (aeval P.val) ((pderiv (P.map i)) (P.relation j))) = ⊤
Module: Mathlib.RingTheory.Extension.Cotangent.LocalizationAway

Goal state before:
pp: case a
R : Type u_1
S : Type u_2
T : Type u_3
ι : Type u_4
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : CommRing T
inst✝³ : Algebra R T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R S T
g : S
inst✝ : IsLocalization.Away g T
P : Generators R S ι
hsple :
  ∀ {x : ((localizationAway T g).comp P).Ring},
    x ∈ Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1} → (compLocalizationAwayAlgHom T g P) x = 0
⊢ Ideal.map ((localizationAway T g).toComp P).toAlgHom P.ker *
        Ideal.map ((localizationAway T g).toComp P).toAlgHom P.ker ⊔
      Ideal.map ((localizationAway T g).toComp P).toAlgHom P.ker *
        Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1} ≤
    RingHom.ker (compLocalizationAwayAlgHom T g P)

pp: case a
R : Type u_1
S : Type u_2
T : Type u_3
ι : Type u_4
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : CommRing T
inst✝³ : Algebra R T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R S T
g : S
inst✝ : IsLocalization.Away g T
P : Generators R S ι
hsple :
  ∀ {x : ((localizationAway T g).comp P).Ring},
    x ∈ Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1} → (compLocalizationAwayAlgHom T g P) x = 0
⊢ Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1} *
        Ideal.map ((localizationAway T g).toComp P).toAlgHom P.ker ⊔
      Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1} *
        Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1} ≤
    RingHom.ker (compLocalizationAwayAlgHom T g P)

Tactic:
·

Goal state after:
case a
R : Type u_1
S : Type u_2
T : Type u_3
ι : Type u_4
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : CommRing T
inst✝³ : Algebra R T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R S T
g : S
inst✝ : IsLocalization.Away g T
P : Generators R S ι
hsple :
  ∀ {x : ((localizationAway T g).comp P).Ring},
    x ∈ Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1} → (compLocalizationAwayAlgHom T g P) x = 0
⊢ Ideal.map ((localizationAway T g).toComp P).toAlgHom P.ker *
        Ideal.map ((localizationAway T g).toComp P).toAlgHom P.ker ⊔
      Ideal.map ((localizationAway T g).toComp P).toAlgHom P.ker *
        Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1} ≤
    RingHom.ker (compLocalizationAwayAlgHom T g P)
Module: Mathlib.RingTheory.Extension.Cotangent.LocalizationAway

Goal state before:
pp: case a.a
R : Type u_1
S : Type u_2
T : Type u_3
ι : Type u_4
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : CommRing T
inst✝³ : Algebra R T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R S T
g : S
inst✝ : IsLocalization.Away g T
P : Generators R S ι
hsple :
  ∀ {x : ((localizationAway T g).comp P).Ring},
    x ∈ Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1} → (compLocalizationAwayAlgHom T g P) x = 0
⊢ Ideal.map ((localizationAway T g).toComp P).toAlgHom P.ker *
      Ideal.map ((localizationAway T g).toComp P).toAlgHom P.ker ≤
    RingHom.ker (compLocalizationAwayAlgHom T g P)

pp: case a
R : Type u_1
S : Type u_2
T : Type u_3
ι : Type u_4
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : CommRing T
inst✝³ : Algebra R T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R S T
g : S
inst✝ : IsLocalization.Away g T
P : Generators R S ι
hsple :
  ∀ {x : ((localizationAway T g).comp P).Ring},
    x ∈ Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1} → (compLocalizationAwayAlgHom T g P) x = 0
⊢ Ideal.map ((localizationAway T g).toComp P).toAlgHom P.ker *
      Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1} ≤
    RingHom.ker (compLocalizationAwayAlgHom T g P)

Tactic:
·

Goal state after:
case a.a
R : Type u_1
S : Type u_2
T : Type u_3
ι : Type u_4
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : CommRing T
inst✝³ : Algebra R T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R S T
g : S
inst✝ : IsLocalization.Away g T
P : Generators R S ι
hsple :
  ∀ {x : ((localizationAway T g).comp P).Ring},
    x ∈ Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1} → (compLocalizationAwayAlgHom T g P) x = 0
⊢ Ideal.map ((localizationAway T g).toComp P).toAlgHom P.ker *
      Ideal.map ((localizationAway T g).toComp P).toAlgHom P.ker ≤
    RingHom.ker (compLocalizationAwayAlgHom T g P)
Module: Mathlib.RingTheory.Extension.Cotangent.LocalizationAway

Goal state before:
pp: case a
R : Type u_1
S : Type u_2
T : Type u_3
ι : Type u_4
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : CommRing T
inst✝³ : Algebra R T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R S T
g : S
inst✝ : IsLocalization.Away g T
P : Generators R S ι
hsple :
  ∀ {x : ((localizationAway T g).comp P).Ring},
    x ∈ Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1} → (compLocalizationAwayAlgHom T g P) x = 0
⊢ Ideal.map ((localizationAway T g).toComp P).toAlgHom P.ker *
      Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1} ≤
    RingHom.ker (compLocalizationAwayAlgHom T g P)

Tactic:
·

Goal state after:
case a
R : Type u_1
S : Type u_2
T : Type u_3
ι : Type u_4
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : CommRing T
inst✝³ : Algebra R T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R S T
g : S
inst✝ : IsLocalization.Away g T
P : Generators R S ι
hsple :
  ∀ {x : ((localizationAway T g).comp P).Ring},
    x ∈ Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1} → (compLocalizationAwayAlgHom T g P) x = 0
⊢ Ideal.map ((localizationAway T g).toComp P).toAlgHom P.ker *
      Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1} ≤
    RingHom.ker (compLocalizationAwayAlgHom T g P)
Module: Mathlib.RingTheory.Extension.Cotangent.LocalizationAway

Goal state before:
pp: case a
R : Type u_1
S : Type u_2
T : Type u_3
ι : Type u_4
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : CommRing T
inst✝³ : Algebra R T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R S T
g : S
inst✝ : IsLocalization.Away g T
P : Generators R S ι
hsple :
  ∀ {x : ((localizationAway T g).comp P).Ring},
    x ∈ Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1} → (compLocalizationAwayAlgHom T g P) x = 0
⊢ Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1} *
        Ideal.map ((localizationAway T g).toComp P).toAlgHom P.ker ⊔
      Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1} *
        Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1} ≤
    RingHom.ker (compLocalizationAwayAlgHom T g P)

Tactic:
·

Goal state after:
case a
R : Type u_1
S : Type u_2
T : Type u_3
ι : Type u_4
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : CommRing T
inst✝³ : Algebra R T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R S T
g : S
inst✝ : IsLocalization.Away g T
P : Generators R S ι
hsple :
  ∀ {x : ((localizationAway T g).comp P).Ring},
    x ∈ Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1} → (compLocalizationAwayAlgHom T g P) x = 0
⊢ Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1} *
        Ideal.map ((localizationAway T g).toComp P).toAlgHom P.ker ⊔
      Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1} *
        Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1} ≤
    RingHom.ker (compLocalizationAwayAlgHom T g P)
Module: Mathlib.RingTheory.Extension.Cotangent.LocalizationAway

Goal state before:
pp: case a.a
R : Type u_1
S : Type u_2
T : Type u_3
ι : Type u_4
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : CommRing T
inst✝³ : Algebra R T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R S T
g : S
inst✝ : IsLocalization.Away g T
P : Generators R S ι
hsple :
  ∀ {x : ((localizationAway T g).comp P).Ring},
    x ∈ Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1} → (compLocalizationAwayAlgHom T g P) x = 0
⊢ Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1} *
      Ideal.map ((localizationAway T g).toComp P).toAlgHom P.ker ≤
    RingHom.ker (compLocalizationAwayAlgHom T g P)

Tactic:
·

Goal state after:
case a.a
R : Type u_1
S : Type u_2
T : Type u_3
ι : Type u_4
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : CommRing T
inst✝³ : Algebra R T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R S T
g : S
inst✝ : IsLocalization.Away g T
P : Generators R S ι
hsple :
  ∀ {x : ((localizationAway T g).comp P).Ring},
    x ∈ Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1} → (compLocalizationAwayAlgHom T g P) x = 0
⊢ Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1} *
      Ideal.map ((localizationAway T g).toComp P).toAlgHom P.ker ≤
    RingHom.ker (compLocalizationAwayAlgHom T g P)
Module: Mathlib.RingTheory.Smooth.StandardSmoothOfFree

Goal state before:
pp: case inr
R : Type u_1
S : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
inst✝² : FinitePresentation R S
p : Ideal S
inst✝¹ : p.IsPrime
inst✝ : IsSmoothAt R p
this :
  ∀ (R : Type u_1) {S : Type u_2} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S]
    [FinitePresentation R S] (p : Ideal S) [inst_4 : p.IsPrime] [IsSmoothAt R p],
    Smooth R S → ∃ f ∉ p, IsStandardSmooth R (Localization.Away f)
h : ¬Smooth R S
⊢ ∃ f ∉ p, IsStandardSmooth R (Localization.Away f)

pp: R✝ : Type u_1
S✝ : Type u_2
inst✝⁸ : CommRing R✝
inst✝⁷ : CommRing S✝
inst✝⁶ : Algebra R✝ S✝
R : Type u_1
S : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
inst✝² : FinitePresentation R S
p : Ideal S
inst✝¹ : p.IsPrime
inst✝ : IsSmoothAt R p
h : Smooth R S
⊢ ∃ f ∉ p, IsStandardSmooth R (Localization.Away f)

Tactic:
·

Goal state after:
case inr
R : Type u_1
S : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
inst✝² : FinitePresentation R S
p : Ideal S
inst✝¹ : p.IsPrime
inst✝ : IsSmoothAt R p
this :
  ∀ (R : Type u_1) {S : Type u_2} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S]
    [FinitePresentation R S] (p : Ideal S) [inst_4 : p.IsPrime] [IsSmoothAt R p],
    Smooth R S → ∃ f ∉ p, IsStandardSmooth R (Localization.Away f)
h : ¬Smooth R S
⊢ ∃ f ∉ p, IsStandardSmooth R (Localization.Away f)
Module: Mathlib.RingTheory.Smooth.StandardSmoothOfFree

Goal state before:
pp: case inr
R : Type u_1
S : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
inst✝² : FinitePresentation R S
p : Ideal S
inst✝¹ : p.IsPrime
inst✝ : IsSmoothAt R p
this :
  ∀ (R : Type u_1) {S : Type u_2} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S]
    [FinitePresentation R S] (p : Ideal S) [inst_4 : p.IsPrime] [IsSmoothAt R p],
    Smooth R S → ∃ f ∉ p, IsStandardSmooth R (Localization.Away f)
h : ¬Smooth R S
⊢ ∃ f ∉ p, IsStandardSmooth R (Localization.Away f)

Tactic:
obtain ⟨g, hg, hsm⟩ := IsSmoothAt.exists_notMem_smooth R p
    have _ : (Ideal.map (algebraMap S (Localization.Away g)) p).IsPrime := by
      apply IsLocalization.isPrime_of_isPrime_disjoint (.powers g) _ _ ‹_›
      rwa [Ideal.disjoint_powers_iff_notMem_of_isPrime]
    obtain ⟨g', hg', hstd⟩ := this (R := R) (p.map (algebraMap S (Localization.Away g))) hsm
    have : IsLocalization.Away (g * (IsLocalization.Away.sec g g').1) (Localization.Away g') :=
      .mul_of_associated _ _ g' <| IsLocalization.Away.associated_sec_fst g g'
    let e : Localization.Away (g * (IsLocalization.Away.sec g g').1) ≃ₐ[S] Localization.Away g' :=
      Localization.algEquiv _ _
    refine ⟨g * (IsLocalization.Away.sec g g').1, ?_, .of_algEquiv (e.restrictScalars R).symm⟩
    refine Ideal.IsPrime.mul_notMem ‹_› hg fun hmem ↦ hg' ?_
    rw [Ideal.mem_iff_of_associated (IsLocalization.Away.associated_sec_fst g g').symm]
    exact Ideal.mem_map_of_mem (algebraMap S (Localization.Away g)) hmem

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

Goal state before:
pp: R✝ : Type u_1
S✝ : Type u_2
inst✝⁸ : CommRing R✝
inst✝⁷ : CommRing S✝
inst✝⁶ : Algebra R✝ S✝
R : Type u_1
S : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
inst✝² : FinitePresentation R S
p : Ideal S
inst✝¹ : p.IsPrime
inst✝ : IsSmoothAt R p
h : Smooth R S
κ : Type u_2
a : κ → S
b : Module.Basis κ (Localization.AtPrime p) Ω[Localization.AtPrime p⁄R]
hb : ∀ (i : κ), b i = (⇑(map R R S (Localization.AtPrime p)) ∘ ⇑(D R S)) (a i)
e : (κ →₀ S) →ₗ[S] Ω[S⁄R] := ⋯
l₁ : (κ →₀ S) →ₗ[S] κ →₀ Localization.AtPrime p := ⋯
l₂ : Ω[S⁄R] →ₗ[S] Ω[Localization.AtPrime p⁄R] := ⋯
eₚ : (κ →₀ Localization.AtPrime p) →ₗ[Localization.AtPrime p] Ω[Localization.AtPrime p⁄R] := ⋯
i : κ
⊢ (eₚ ∘ₗ Finsupp.lsingle i) 1 = ((IsLocalizedModule.map p.primeCompl l₁ l₂) e) (l₁ (Finsupp.single i 1))

pp: R✝ : Type u_1
S✝ : Type u_2
inst✝⁸ : CommRing R✝
inst✝⁷ : CommRing S✝
inst✝⁶ : Algebra R✝ S✝
R : Type u_1
S : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
inst✝² : FinitePresentation R S
p : Ideal S
inst✝¹ : p.IsPrime
inst✝ : IsSmoothAt R p
h : Smooth R S
κ : Type u_2
a : κ → S
b : Module.Basis κ (Localization.AtPrime p) Ω[Localization.AtPrime p⁄R]
hb : ∀ (i : κ), b i = (⇑(map R R S (Localization.AtPrime p)) ∘ ⇑(D R S)) (a i)
e : (κ →₀ S) →ₗ[S] Ω[S⁄R] := ⋯
l₁ : (κ →₀ S) →ₗ[S] κ →₀ Localization.AtPrime p := ⋯
l₂ : Ω[S⁄R] →ₗ[S] Ω[Localization.AtPrime p⁄R] := ⋯
eₚ : (κ →₀ Localization.AtPrime p) →ₗ[Localization.AtPrime p] Ω[Localization.AtPrime p⁄R] := ⋯
i : κ
⊢ ((IsLocalizedModule.map p.primeCompl l₁ l₂) e) (l₁ (Finsupp.single i 1)) = (↑b.repr.symm ∘ₗ Finsupp.lsingle i) 1

Tactic:
·

Goal state after:
R✝ : Type u_1
S✝ : Type u_2
inst✝⁸ : CommRing R✝
inst✝⁷ : CommRing S✝
inst✝⁶ : Algebra R✝ S✝
R : Type u_1
S : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
inst✝² : FinitePresentation R S
p : Ideal S
inst✝¹ : p.IsPrime
inst✝ : IsSmoothAt R p
h : Smooth R S
κ : Type u_2
a : κ → S
b : Module.Basis κ (Localization.AtPrime p) Ω[Localization.AtPrime p⁄R]
hb : ∀ (i : κ), b i = (⇑(map R R S (Localization.AtPrime p)) ∘ ⇑(D R S)) (a i)
e : (κ →₀ S) →ₗ[S] Ω[S⁄R] := ⋯
l₁ : (κ →₀ S) →ₗ[S] κ →₀ Localization.AtPrime p := ⋯
l₂ : Ω[S⁄R] →ₗ[S] Ω[Localization.AtPrime p⁄R] := ⋯
eₚ : (κ →₀ Localization.AtPrime p) →ₗ[Localization.AtPrime p] Ω[Localization.AtPrime p⁄R] := ⋯
i : κ
⊢ (eₚ ∘ₗ Finsupp.lsingle i) 1 = ((IsLocalizedModule.map p.primeCompl l₁ l₂) e) (l₁ (Finsupp.single i 1))
Module: Mathlib.RingTheory.Extension.Cotangent.LocalizationAway

Goal state before:
pp: case a
R : Type u_1
S : Type u_2
T : Type u_3
ι : Type u_4
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : CommRing T
inst✝³ : Algebra R T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R S T
g : S
inst✝ : IsLocalization.Away g T
P : Generators R S ι
hsple :
  ∀ {x : ((localizationAway T g).comp P).Ring},
    x ∈ Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1} → (compLocalizationAwayAlgHom T g P) x = 0
⊢ Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1} *
      Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1} ≤
    RingHom.ker (compLocalizationAwayAlgHom T g P)

Tactic:
·

Goal state after:
case a
R : Type u_1
S : Type u_2
T : Type u_3
ι : Type u_4
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : CommRing T
inst✝³ : Algebra R T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R S T
g : S
inst✝ : IsLocalization.Away g T
P : Generators R S ι
hsple :
  ∀ {x : ((localizationAway T g).comp P).Ring},
    x ∈ Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1} → (compLocalizationAwayAlgHom T g P) x = 0
⊢ Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1} *
      Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1} ≤
    RingHom.ker (compLocalizationAwayAlgHom T g P)
Module: Mathlib.RingTheory.Smooth.StandardSmoothOfFree

Goal state before:
pp: R✝ : Type u_1
S✝ : Type u_2
inst✝⁸ : CommRing R✝
inst✝⁷ : CommRing S✝
inst✝⁶ : Algebra R✝ S✝
R : Type u_1
S : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
inst✝² : FinitePresentation R S
p : Ideal S
inst✝¹ : p.IsPrime
inst✝ : IsSmoothAt R p
h : Smooth R S
κ : Type u_2
a : κ → S
b : Module.Basis κ (Localization.AtPrime p) Ω[Localization.AtPrime p⁄R]
hb : ∀ (i : κ), b i = (⇑(map R R S (Localization.AtPrime p)) ∘ ⇑(D R S)) (a i)
e : (κ →₀ S) →ₗ[S] Ω[S⁄R] := ⋯
l₁ : (κ →₀ S) →ₗ[S] κ →₀ Localization.AtPrime p := ⋯
l₂ : Ω[S⁄R] →ₗ[S] Ω[Localization.AtPrime p⁄R] := ⋯
eₚ : (κ →₀ Localization.AtPrime p) →ₗ[Localization.AtPrime p] Ω[Localization.AtPrime p⁄R] := ⋯
i : κ
⊢ ((IsLocalizedModule.map p.primeCompl l₁ l₂) e) (l₁ (Finsupp.single i 1)) = (↑b.repr.symm ∘ₗ Finsupp.lsingle i) 1

Tactic:
·

Goal state after:
R✝ : Type u_1
S✝ : Type u_2
inst✝⁸ : CommRing R✝
inst✝⁷ : CommRing S✝
inst✝⁶ : Algebra R✝ S✝
R : Type u_1
S : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
inst✝² : FinitePresentation R S
p : Ideal S
inst✝¹ : p.IsPrime
inst✝ : IsSmoothAt R p
h : Smooth R S
κ : Type u_2
a : κ → S
b : Module.Basis κ (Localization.AtPrime p) Ω[Localization.AtPrime p⁄R]
hb : ∀ (i : κ), b i = (⇑(map R R S (Localization.AtPrime p)) ∘ ⇑(D R S)) (a i)
e : (κ →₀ S) →ₗ[S] Ω[S⁄R] := ⋯
l₁ : (κ →₀ S) →ₗ[S] κ →₀ Localization.AtPrime p := ⋯
l₂ : Ω[S⁄R] →ₗ[S] Ω[Localization.AtPrime p⁄R] := ⋯
eₚ : (κ →₀ Localization.AtPrime p) →ₗ[Localization.AtPrime p] Ω[Localization.AtPrime p⁄R] := ⋯
i : κ
⊢ ((IsLocalizedModule.map p.primeCompl l₁ l₂) e) (l₁ (Finsupp.single i 1)) = (↑b.repr.symm ∘ₗ Finsupp.lsingle i) 1
Module: Mathlib.RingTheory.Smooth.StandardSmoothOfFree

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : Smooth R S
⊢ ∃ s, Ideal.span s = ⊤ ∧ ∀ x ∈ s, IsStandardSmooth R (Localization.Away x)

Tactic:
choose f hf₁ hf₂ using IsSmoothAt.exists_notMem_isStandardSmooth R (S := S)

Goal state after:
R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : Smooth R S
f : (p : Ideal S) → [inst : p.IsPrime] → [IsSmoothAt R p] → S
hf₁ : ∀ (p : Ideal S) [inst : p.IsPrime] [inst_1 : IsSmoothAt R p], f p ∉ p
hf₂ : ∀ (p : Ideal S) [inst : p.IsPrime] [inst_1 : IsSmoothAt R p], IsStandardSmooth R (Localization.Away (f p))
⊢ ∃ s, Ideal.span s = ⊤ ∧ ∀ x ∈ s, IsStandardSmooth R (Localization.Away x)
Module: Mathlib.RingTheory.Extension.Cotangent.Basis

Goal state before:
pp: case refine_1
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
σ✝ : Type u_3
ι : Type u_4
P : Generators R S ι
σ : Type u_5
b : Module.Basis σ S P.toExtension.Cotangent
D : Aux P b
⊢ Function.Surjective ⇑(algebraMap D.T S)

pp: case refine_2
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
σ✝ : Type u_3
ι : Type u_4
P : Generators R S ι
σ : Type u_5
b : Module.Basis σ S P.toExtension.Cotangent
D : Aux P b
⊢ Function.Surjective ⇑(algebraMap P.Ring D.T)

pp: case refine_3
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
σ✝ : Type u_3
ι : Type u_4
P : Generators R S ι
σ : Type u_5
b : Module.Basis σ S P.toExtension.Cotangent
D : Aux P b
⊢ IsUnit ((algebraMap P.Ring S) D.g)

Tactic:
·

Goal state after:
case refine_1
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
σ✝ : Type u_3
ι : Type u_4
P : Generators R S ι
σ : Type u_5
b : Module.Basis σ S P.toExtension.Cotangent
D : Aux P b
⊢ Function.Surjective ⇑(algebraMap D.T S)
Module: Mathlib.RingTheory.Extension.Cotangent.Basis

Goal state before:
pp: case refine_2
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
σ✝ : Type u_3
ι : Type u_4
P : Generators R S ι
σ : Type u_5
b : Module.Basis σ S P.toExtension.Cotangent
D : Aux P b
⊢ Function.Surjective ⇑(algebraMap P.Ring D.T)

pp: case refine_3
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
σ✝ : Type u_3
ι : Type u_4
P : Generators R S ι
σ : Type u_5
b : Module.Basis σ S P.toExtension.Cotangent
D : Aux P b
⊢ IsUnit ((algebraMap P.Ring S) D.g)

Tactic:
·

Goal state after:
case refine_2
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
σ✝ : Type u_3
ι : Type u_4
P : Generators R S ι
σ : Type u_5
b : Module.Basis σ S P.toExtension.Cotangent
D : Aux P b
⊢ Function.Surjective ⇑(algebraMap P.Ring D.T)
Module: Mathlib.RingTheory.Extension.Cotangent.Basis

Goal state before:
pp: case refine_3
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
σ✝ : Type u_3
ι : Type u_4
P : Generators R S ι
σ : Type u_5
b : Module.Basis σ S P.toExtension.Cotangent
D : Aux P b
⊢ IsUnit ((algebraMap P.Ring S) D.g)

Tactic:
·

Goal state after:
case refine_3
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
σ✝ : Type u_3
ι : Type u_4
P : Generators R S ι
σ : Type u_5
b : Module.Basis σ S P.toExtension.Cotangent
D : Aux P b
⊢ IsUnit ((algebraMap P.Ring S) D.g)
Module: Mathlib.RingTheory.Extension.Cotangent.Basis

Goal state before:
pp: case pos
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
σ✝ : Type u_3
ι : Type u_4
P : Generators R S ι
σ : Type u_5
b : Module.Basis σ S P.toExtension.Cotangent
D : Aux P b
x : MvPolynomial ι R ⧸ Ideal.span (Set.range (Subtype.val ∘ D.f ∘ ⇑b))
h✝ : x = 0
⊢ (Ideal.Quotient.mk (Ideal.span (Set.range (Subtype.val ∘ D.f ∘ ⇑b)))) 0 = x

pp: case pos
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
σ✝ : Type u_3
ι : Type u_4
P : Generators R S ι
σ : Type u_5
b : Module.Basis σ S P.toExtension.Cotangent
D : Aux P b
x : MvPolynomial ι R ⧸ Ideal.span (Set.range (Subtype.val ∘ D.f ∘ ⇑b))
h✝¹ : ¬x = 0
h✝ : x = -1
⊢ (Ideal.Quotient.mk (Ideal.span (Set.range (Subtype.val ∘ D.f ∘ ⇑b)))) (-1) = x

pp: case neg
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
σ✝ : Type u_3
ι : Type u_4
P : Generators R S ι
σ : Type u_5
b : Module.Basis σ S P.toExtension.Cotangent
D : Aux P b
x : MvPolynomial ι R ⧸ Ideal.span (Set.range (Subtype.val ∘ D.f ∘ ⇑b))
h✝¹ : ¬x = 0
h✝ : ¬x = -1
⊢ (Ideal.Quotient.mk (Ideal.span (Set.range (Subtype.val ∘ D.f ∘ ⇑b)))) (Function.surjInv ⋯ x) = x

Tactic:
·

Goal state after:
case pos
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
σ✝ : Type u_3
ι : Type u_4
P : Generators R S ι
σ : Type u_5
b : Module.Basis σ S P.toExtension.Cotangent
D : Aux P b
x : MvPolynomial ι R ⧸ Ideal.span (Set.range (Subtype.val ∘ D.f ∘ ⇑b))
h✝ : x = 0
⊢ (Ideal.Quotient.mk (Ideal.span (Set.range (Subtype.val ∘ D.f ∘ ⇑b)))) 0 = x