mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.Unramified.Finite

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
inst✝² : FormallyUnramified R S
inst✝¹ : EssFiniteType R S
inst✝ : Free R S
I : Type u_2 := Free.ChooseBasisIndex R S
b : Basis I R S := Free.chooseBasis R S
f : I →₀ S
hf : elem R S = f.sum fun i x ↦ x ⊗ₜ[R] b i
x : S
a : I → I →₀ R := fun i ↦ b.repr (b i * x)
F : I →₀ I →₀ R := Finsupp.onFinset f.support (fun j ↦ b.repr (x * f j)) ⋯
hG : ∀ j ∉ f.support.biUnion fun i ↦ (a i).support, b.repr (f.sum fun i y ↦ (a i) j • y) = 0
G : I →₀ I →₀ R :=
  Finsupp.onFinset (f.support.biUnion fun i ↦ (a i).support) (fun j ↦ b.repr (f.sum fun i y ↦ (a i) j • y)) ⋯
this : F = G
⊢ ∀ (j : I), x * f j = f.sum fun i y ↦ (a i) j • y

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.QuasiFinite.Basic

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : IsArtinianRing R
p : PrimeSpectrum S
inst✝ : FiniteType R S
H✝ : IsClopen {p}
this✝² : IsNoetherianRing S
this✝¹ : IsJacobsonRing S
e : S
he : IsIdempotentElem e
H : {p} = ↑(PrimeSpectrum.basicOpen e)
hep : e ∉ p.asIdeal
f : Localization.Away e →ₐ[S] Localization.AtPrime p.asIdeal := IsLocalization.Away.liftAlgHom e ⋯
h₁ : Set.range (PrimeSpectrum.comap (algebraMap S (Localization.Away e))) = {p}
this✝ : Subsingleton (PrimeSpectrum (Localization.Away e))
x : S
s : ↥p.asIdeal.primeCompl
this : IsUnit ((algebraMap S (Localization.Away e)) ↑s)
⊢ ∃ a, f a = IsLocalization.mk' (Localization.AtPrime p.asIdeal) x s

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.QuasiFinite.Basic

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : IsArtinianRing R
p : PrimeSpectrum S
inst✝ : FiniteType R S
H✝¹ : IsClopen {p}
this✝¹ : IsNoetherianRing S
this✝ : IsJacobsonRing S
e : S
he : IsIdempotentElem e
H✝ : {p} = ↑(PrimeSpectrum.basicOpen e)
hep : e ∉ p.asIdeal
f : Localization.Away e →ₐ[S] Localization.AtPrime p.asIdeal := IsLocalization.Away.liftAlgHom e ⋯
h₁ : Set.range (PrimeSpectrum.comap (algebraMap S (Localization.Away e))) = {p}
this : Subsingleton (PrimeSpectrum (Localization.Away e))
x : S
s : ↥p.asIdeal.primeCompl
H : ¬IsUnit ((algebraMap S (Localization.Away e)) ↑s)
⊢ Ideal.span {(algebraMap S (Localization.Away e)) ↑s} ≠ ⊤

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.QuasiFinite.Basic

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : IsArtinianRing R
p : PrimeSpectrum S
inst✝ : FiniteType R S
H✝ : IsClopen {p}
this✝¹ : IsNoetherianRing S
this✝ : IsJacobsonRing S
e : S
he : IsIdempotentElem e
H : {p} = ↑(PrimeSpectrum.basicOpen e)
hep : e ∉ p.asIdeal
f : Localization.Away e →ₐ[S] Localization.AtPrime p.asIdeal := IsLocalization.Away.liftAlgHom e ⋯
h₁ : Set.range (PrimeSpectrum.comap (algebraMap S (Localization.Away e))) = {p}
this : Subsingleton (PrimeSpectrum (Localization.Away e))
⊢ Function.Surjective ⇑f

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.QuasiFinite.Basic

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : IsArtinianRing R
p : PrimeSpectrum S
inst✝ : FiniteType R S
H : IsOpen {p}
⊢ QuasiFiniteAt R p.asIdeal

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.QuasiFinite.Basic

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
p : Ideal R
inst✝⁴ : p.IsPrime
q : Ideal S
inst✝³ : q.IsPrime
inst✝² : q.LiesOver p
inst✝¹ : EssFiniteType R S
inst✝ : QuasiFiniteAt R q
e : ↑(PrimeSpectrum.comap (algebraMap R S) ⁻¹' {{ asIdeal := p, isPrime := ⋯ }}) ≃ₜ
  PrimeSpectrum ({ asIdeal := p, isPrime := ⋯ }.asIdeal.Fiber S) :=
  PrimeSpectrum.preimageHomeomorphFiber R S { asIdeal := p, isPrime := ⋯ }
qF : PrimeSpectrum (p.Fiber S) := e ⟨{ asIdeal := q, isPrime := inst✝³ }, ⋯⟩
this : QuasiFiniteAt p.ResidueField qF.asIdeal
r : p.Fiber S
hr : r ∉ qF.asIdeal
hrq : ↑(PrimeSpectrum.basicOpen r) = {{ asIdeal := qF.asIdeal, isPrime := ⋯ }}
s : R
hs : s ∉ p
x : S
hsx : s • r = 1 ⊗ₜ[R] x
⊢ x ∉ q

Tactic:
by

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

Goal state before:
pp: R : Type u
A : Type v
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Algebra R A
B : Type u_1
inst✝² : CommRing B
inst✝¹ : Algebra R B
inst✝ : FormallySmooth R A
I : Ideal B
hI : IsNilpotent I
g : A →ₐ[R] B ⧸ I
⊢ ∃ f, (Ideal.Quotient.mkₐ R I).comp f = g

Tactic:
by

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

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
inst✝² : FormallyUnramified R S
inst✝¹ : EssFiniteType R S
inst✝ : Free R S
⊢ Module.Finite R S

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.QuasiFinite.Basic

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
p : Ideal R
inst✝⁴ : p.IsPrime
q : Ideal S
inst✝³ : q.IsPrime
inst✝² : q.LiesOver p
inst✝¹ : EssFiniteType R S
inst✝ : QuasiFiniteAt R q
⊢ ∃ s ∉ q, ∀ (q' : Ideal S), q'.IsPrime → q' ≠ q → q'.LiesOver p → s ∈ q'

Tactic:
by

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

Goal state before:
pp: R : Type u
A : Type v
inst✝⁷ : CommRing R
inst✝⁶ : CommRing A
inst✝⁵ : Algebra R A
B : Type u_1
inst✝⁴ : CommRing B
inst✝³ : Algebra R B
C : Type u_4
inst✝² : CommRing C
inst✝¹ : Algebra R C
inst✝ : FormallySmooth R A
f : A →ₐ[R] C
g : B →ₐ[R] C
hg : Function.Surjective ⇑g
hg' : IsNilpotent (RingHom.ker g)
x : A
⊢ g ((liftOfSurjective f g hg hg') x) = f x

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.QuasiFinite.Basic

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁹ : CommRing R
inst✝⁸ : CommRing S
inst✝⁷ : Algebra R S
inst✝⁶ : FiniteType R S
p : Ideal R
inst✝⁵ : p.IsPrime
q : Ideal S
inst✝⁴ : q.IsPrime
inst✝³ : q.LiesOver p
inst✝² : QuasiFiniteAt R q
inst✝¹ : Algebra (Localization.AtPrime p) (Localization.AtPrime q)
inst✝ : Localization.AtPrime.IsLiesOverAlgebra p q
⊢ Function.Surjective
    ⇑(TensorProduct.lift (ofId p.ResidueField q.ResidueField) (IsScalarTower.toAlgHom R S q.ResidueField) ⋯)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Etale.Basic

Goal state before:
pp: R : Type u
A : Type v
inst✝² : CommRing R
inst✝¹ : CommRing A
inst✝ : Algebra R A
H :
  ∀ ⦃B : Type (max u v)⦄ [inst : CommRing B] [inst_1 : Algebra R B] (I : Ideal B),
    I ^ 2 = ⊥ → Function.Bijective (Ideal.Quotient.mkₐ R I).comp
⊢ ∀ ⦃B : Type (max u v)⦄ [inst : CommRing B] [inst_1 : Algebra R B] (I : Ideal B),
    I ^ 2 = ⊥ → Function.Injective (Ideal.Quotient.mkₐ R I).comp

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Etale.Basic

Goal state before:
pp: R : Type u
A : Type v
inst✝² : CommRing R
inst✝¹ : CommRing A
inst✝ : Algebra R A
H :
  ∀ ⦃B : Type (max u v)⦄ [inst : CommRing B] [inst_1 : Algebra R B] (I : Ideal B),
    I ^ 2 = ⊥ → Function.Bijective (Ideal.Quotient.mkₐ R I).comp
this : FormallyUnramified R A
⊢ ∀ ⦃B : Type (max u v)⦄ [inst : CommRing B] [inst_1 : Algebra R B] (I : Ideal B),
    I ^ 2 = ⊥ → Function.Surjective (Ideal.Quotient.mkₐ R I).comp

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Etale.Basic

Goal state before:
pp: R : Type u_2
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
h : Function.Surjective ⇑(algebraMap R S)
⊢ FormallyEtale R S ↔ IsIdempotentElem (RingHom.ker (algebraMap R S))

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Etale.Basic

Goal state before:
pp: R : Type u_2
S : Type u_3
Rₘ : Type u_4
Sₘ : Type u_5
inst✝¹³ : CommRing R
inst✝¹² : CommRing S
inst✝¹¹ : CommRing Rₘ
inst✝¹⁰ : CommRing Sₘ
M : Submonoid R
inst✝⁹ : Algebra R S
inst✝⁸ : Algebra R Sₘ
inst✝⁷ : Algebra S Sₘ
inst✝⁶ : Algebra R Rₘ
inst✝⁵ : Algebra Rₘ Sₘ
inst✝⁴ : IsScalarTower R Rₘ Sₘ
inst✝³ : IsScalarTower R S Sₘ
inst✝² : IsLocalization M Rₘ
inst✝¹ : IsLocalization (Submonoid.map (algebraMap R S) M) Sₘ
inst✝ : FormallyEtale R S
⊢ FormallyEtale Rₘ Sₘ

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Etale.Basic

Goal state before:
pp: R : Type u
A : Type v
inst✝² : CommRing R
inst✝¹ : CommRing A
inst✝ : Algebra R A
⊢ Etale R A ↔ FormallyUnramified R A ∧ Smooth R A

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Etale.Basic

Goal state before:
pp: R✝ : Type u
A : Type v
B : Type u_1
inst✝⁶ : CommRing R✝
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R✝ A
inst✝³ : CommRing B
inst✝² : Algebra R✝ B
R S : Type u
inst✝¹ : CommRing R
inst✝ : CommRing S
⊢ Etale (R × S) S

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Etale.Basic

Goal state before:
pp: R : Type u
A : Type v
B : Type u_1
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Algebra R A
inst✝² : CommRing B
inst✝¹ : Algebra R B
S : Type u_2
inst✝ : CommRing S
⊢ Etale (R × S) R

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Etale.Basic

Goal state before:
pp: R : Type u
A : Type v
B : Type u_1
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Algebra R A
inst✝² : CommRing B
inst✝¹ : Algebra R B
S : Type u_2
inst✝ : CommRing S
⊢ Etale (R × S) S

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Etale.Basic

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
⊢ (algebraMap R S).FormallyEtale ↔ Algebra.FormallyEtale R S

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Etale.Basic

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
T : Type u_3
inst✝ : CommRing T
f : R →+* S
g : S →+* T
hf : f.FormallyEtale
hg : g.FormallyEtale
⊢ (g.comp f).FormallyEtale

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Kaehler.JacobiZariski

Goal state before:
pp: R : Type u₁
S : Type u₂
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
T : Type u₃
inst✝³ : CommRing T
inst✝² : Algebra R T
inst✝¹ : Algebra S T
inst✝ : IsScalarTower R S T
ι : Type w₁
σ : Type w₂
Q : Generators S T ι
P : Generators R S σ
⊢ Function.Surjective ⇑(Extension.Cotangent.map (Q.ofComp P).toExtensionHom)

Tactic:
by

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

Goal state before:
pp: R : Type u
A : Type v
inst✝⁹ : CommRing R
inst✝⁸ : CommRing A
inst✝⁷ : Algebra R A
B : Type u_1
P : Type u_2
C : Type u_3
inst✝⁶ : CommRing B
inst✝⁵ : Algebra R B
inst✝⁴ : CommRing C
inst✝³ : Algebra R C
inst✝² : CommRing P
inst✝¹ : Algebra R P
P₁ : Extension R A
P₂ : Extension R A
inst✝ : FormallySmooth R P₁.Ring
lift : P₁.Ring →ₐ[R] P₂.infinitesimal.Ring :=
  FormallySmooth.liftOfSurjective (IsScalarTower.toAlgHom R P₁.Ring A)
    (IsScalarTower.toAlgHom R P₂.infinitesimal.Ring A) ⋯ ⋯
⊢ ∀ a ∈ P₁.ker ^ 2, lift a = 0

Tactic:
by

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

Goal state before:
pp: R : Type u
A : Type v
inst✝⁹ : CommRing R
inst✝⁸ : CommRing A
inst✝⁷ : Algebra R A
B : Type u_1
P : Type u_2
C : Type u_3
inst✝⁶ : CommRing B
inst✝⁵ : Algebra R B
inst✝⁴ : CommRing C
inst✝³ : Algebra R C
inst✝² : CommRing P
inst✝¹ : Algebra R P
P₁ : Extension R A
P₂ : Extension R A
inst✝ : FormallySmooth R P₁.Ring
lift : P₁.Ring →ₐ[R] P₂.infinitesimal.Ring :=
  FormallySmooth.liftOfSurjective (IsScalarTower.toAlgHom R P₁.Ring A)
    (IsScalarTower.toAlgHom R P₂.infinitesimal.Ring A) ⋯ ⋯
⊢ ∀ (x : R),
    (Ideal.Quotient.liftₐ (P₁.ker ^ 2) lift ⋯).toRingHom ((algebraMap R P₁.infinitesimal.Ring) x) =
      (algebraMap R P₂.infinitesimal.Ring) ((algebraMap R R) x)

Tactic:
by

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

Goal state before:
pp: R : Type u
A : Type v
inst✝⁹ : CommRing R
inst✝⁸ : CommRing A
inst✝⁷ : Algebra R A
B : Type u_1
P : Type u_2
C : Type u_3
inst✝⁶ : CommRing B
inst✝⁵ : Algebra R B
inst✝⁴ : CommRing C
inst✝³ : Algebra R C
inst✝² : CommRing P
inst✝¹ : Algebra R P
P₁ : Extension R A
P₂ : Extension R A
inst✝ : FormallySmooth R P₁.Ring
lift : P₁.Ring →ₐ[R] P₂.infinitesimal.Ring :=
  FormallySmooth.liftOfSurjective (IsScalarTower.toAlgHom R P₁.Ring A)
    (IsScalarTower.toAlgHom R P₂.infinitesimal.Ring A) ⋯ ⋯
x : P₁.infinitesimal.Ring
⊢ (algebraMap P₂.infinitesimal.Ring A) ((Ideal.Quotient.liftₐ (P₁.ker ^ 2) lift ⋯).toRingHom x) =
    (algebraMap A A) ((algebraMap P₁.infinitesimal.Ring A) x)

Tactic:
by

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

Goal state before:
pp: R : Type u
A : Type v
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
P₁ : Extension R A
P₂ : Extension R A
f : P₁.Hom P₂
inst✝¹ : FormallySmooth R P₁.Ring
inst✝ : FormallySmooth R P₂.Ring
⊢ ↑(equivOfFormallySmooth P₁ P₂) = map f

Tactic:
by

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

Goal state before:
pp: R : Type u
A : Type v
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
P₁ : Extension R A
P₂ : Extension R A
f : P₁.Hom P₂
inst✝¹ : FormallySmooth R P₁.Ring
inst✝ : FormallySmooth R P₂.Ring
x : P₁.H1Cotangent
⊢ (equivOfFormallySmooth P₁ P₂) x = (map f) x

Tactic:
by

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

Goal state before:
pp: R : Type u
A : Type v
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
P : Extension R A
inst✝ : FormallySmooth R P.Ring
⊢ Function.Injective ⇑P.cotangentComplex ↔ Subsingleton (H1Cotangent R A)

Tactic:
by

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

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
M : Type u_3
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : Module S M
inst✝² : IsScalarTower R S M
inst✝¹ : FormallyUnramified R S
inst✝ : EssFiniteType R S
r : S
m : M
⊢ (((TensorProduct.AlgebraTensorModule.mapBilinear R S S S S S M) LinearMap.id).flip (elem R S) ∘ₗ
          (lsmul R R M).toLinearMap.flip).toFun
      (r • m) =
    (RingHom.id S) r •
      (((TensorProduct.AlgebraTensorModule.mapBilinear R S S S S S M) LinearMap.id).flip (elem R S) ∘ₗ
            (lsmul R R M).toLinearMap.flip).toFun
        m

Tactic:
by

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

Goal state before:
pp: R : Type u
A : Type v
inst✝⁷ : CommRing R
inst✝⁶ : CommRing A
inst✝⁵ : Algebra R A
P : Type u_2
inst✝⁴ : CommRing P
inst✝³ : Algebra R P
inst✝² : FormallySmooth R P
inst✝¹ : Algebra P A
inst✝ : IsScalarTower R P A
hf : Function.Surjective ⇑(algebraMap P A)
⊢ Function.Surjective ⇑(↑P (mapBaseChange R P A))

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Kaehler.JacobiZariski

Goal state before:
pp: R : Type u₁
S : Type u₂
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
T : Type u₃
inst✝³ : CommRing T
inst✝² : Algebra R T
inst✝¹ : Algebra S T
inst✝ : IsScalarTower R S T
ι : Type w₁
σ : Type w₂
Q : Generators S T ι
P : Generators R S σ
x : (Q.comp P).Ring
hx' : x ∈ (Q.comp P).toExtension.ker
hx : (Extension.Cotangent.map (Q.ofComp P).toExtensionHom) (Extension.Cotangent.mk ⟨x, hx'⟩) = 0
⊢ (Q.ofComp P).toAlgHom x ∈ Q.ker ^ 2

Tactic:
by

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

Goal state before:
pp: R : Type u
A : Type v
inst✝⁷ : CommRing R
inst✝⁶ : CommRing A
inst✝⁵ : Algebra R A
P : Type u_2
inst✝⁴ : CommRing P
inst✝³ : Algebra R P
inst✝² : FormallySmooth R P
inst✝¹ : Algebra P A
inst✝ : IsScalarTower R P A
hf : Function.Surjective ⇑(algebraMap P A)
⊢ FormallySmooth R A ↔ ∃ l, l ∘ₗ kerCotangentToTensor R P A = LinearMap.id

Tactic:
by

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

Goal state before:
pp: R : Type u
A : Type v
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
P : Extension R A
inst✝ : FormallySmooth R P.Ring
r : P.Ring
m : P.ker.Cotangent
⊢ (AddEquiv.refl P.ker.Cotangent).toFun (r • m) = (RingHom.id P.Ring) r • (AddEquiv.refl P.ker.Cotangent).toFun m

Tactic:
by

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

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
M : Type u_3
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : Module S M
inst✝² : IsScalarTower R S M
inst✝¹ : FormallyUnramified R S
inst✝ : EssFiniteType R S
⊢ TensorProduct.AlgebraTensorModule.lift (↑R (lsmul S S M).toLinearMap.flip).flip ∘ₗ sec R S M = LinearMap.id

Tactic:
by

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

Goal state before:
pp: R : Type u
A : Type v
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
P : Extension R A
inst✝ : FormallySmooth R P.Ring
⊢ FormallySmooth R A ↔ ∃ l, l ∘ₗ P.cotangentComplex = LinearMap.id

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Kaehler.JacobiZariski

Goal state before:
pp: R : Type u₁
S : Type u₂
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
T : Type u₃
inst✝³ : CommRing T
inst✝² : Algebra R T
inst✝¹ : Algebra S T
inst✝ : IsScalarTower R S T
ι : Type w₁
σ : Type w₂
Q : Generators S T ι
P : Generators R S σ
x : (Q.comp P).Ring
hx' : x ∈ (Q.comp P).toExtension.ker
y : (Q.comp P).Ring
hy : y ∈ (Q.comp P).ker * (Q.comp P).ker
e : x - y ∈ Ideal.map (Q.toComp P).toAlgHom P.ker
z : ↥(Q.comp P).ker := ⟨x - y, ⋯⟩
hz : ↑z ∈ Ideal.map (Q.toComp P).toAlgHom.toRingHom P.ker
⊢ Extension.Cotangent.mk ⟨x, hx'⟩ = Extension.Cotangent.mk z

Tactic:
by

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

Goal state before:
pp: R : Type u
A : Type v
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Algebra R A
P : Type u_2
inst✝² : CommRing P
inst✝¹ : Algebra R P
inst✝ : FormallySmooth R P
f : P →ₐ[R] A
hf : Function.Surjective ⇑f
⊢ FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A

Tactic:
by

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

Goal state before:
pp: R : Type u
A : Type v
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Algebra R A
P : Type u_2
inst✝² : CommRing P
inst✝¹ : Algebra R P
inst✝ : FormallySmooth R P
f : P →ₐ[R] A
g : A →ₐ[R] P ⧸ RingHom.ker f.toRingHom ^ 2
h : f.kerSquareLift.comp g = AlgHom.id R A
⊢ FormallySmooth R A

Tactic:
by

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

Goal state before:
pp: R : Type u
A : Type v
inst✝² : CommRing R
inst✝¹ : CommRing A
inst✝ : Algebra R A
H :
  ∀ ⦃B : Type (max u v)⦄ [inst : CommRing B] [inst_1 : Algebra R B] (I : Ideal B),
    I ^ 2 = ⊥ → Function.Surjective (Ideal.Quotient.mkₐ R I).comp
P : Generators R A A := Generators.self R A
f : P.Ring →ₐ[R] A := IsScalarTower.toAlgHom R P.Ring A
surj : Function.Surjective ⇑f.kerSquareLift
⊢ RingHom.ker f.kerSquareLift.toRingHom ^ 2 = ⊥

Tactic:
by

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

Goal state before:
pp: R : Type u
A : Type v
inst✝² : CommRing R
inst✝¹ : CommRing A
inst✝ : Algebra R A
H :
  ∀ ⦃B : Type (max u v)⦄ [inst : CommRing B] [inst_1 : Algebra R B] (I : Ideal B),
    I ^ 2 = ⊥ → Function.Surjective (Ideal.Quotient.mkₐ R I).comp
⊢ FormallySmooth R A

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Kaehler.JacobiZariski

Goal state before:
pp: R : Type u₁
S : Type u₂
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
T : Type u₃
inst✝³ : CommRing T
inst✝² : Algebra R T
inst✝¹ : Algebra S T
inst✝ : IsScalarTower R S T
ι : Type w₁
σ : Type w₂
Q : Generators S T ι
P : Generators R S σ
⊢ Function.Exact ⇑(LinearMap.liftBaseChange T (Extension.Cotangent.map (Q.toComp P).toExtensionHom))
    ⇑(Extension.Cotangent.map (Q.ofComp P).toExtensionHom)

Tactic:
by

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

Goal state before:
pp: R : Type u_4
inst✝⁸ : CommRing R
A : Type u_5
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
B : Type u_6
inst✝⁵ : CommRing B
inst✝⁴ : Algebra R B
inst✝³ : Algebra A B
inst✝² : IsScalarTower R A B
inst✝¹ : FormallySmooth R A
inst✝ : FormallySmooth A B
⊢ FormallySmooth R B

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Kaehler.JacobiZariski

Goal state before:
pp: R : Type u₁
S : Type u₂
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
T : Type u₃
inst✝³ : CommRing T
inst✝² : Algebra R T
inst✝¹ : Algebra S T
inst✝ : IsScalarTower R S T
ι : Type w₁
σ : Type w₂
Q : Generators S T ι
P : Generators R S σ
⊢ ↑(compEquiv Q P).symm ∘ₗ LinearMap.inr T Q.toExtension.CotangentSpace (T ⊗[S] P.toExtension.CotangentSpace) =
    LinearMap.liftBaseChange T (Extension.CotangentSpace.map (Q.toComp P).toExtensionHom)

Tactic:
by

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

Goal state before:
pp: R : Type u_4
inst✝⁸ : CommRing R
A : Type u_5
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
B : Type u_6
inst✝⁵ : CommRing B
inst✝⁴ : Algebra R B
inst✝³ : Algebra A B
inst✝² : IsScalarTower R A B
inst✝¹ : FormallyUnramified R A
inst✝ : FormallySmooth R B
⊢ FormallySmooth A B

Tactic:
by

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

Goal state before:
pp: R : Type u_4
inst✝² : CommRing R
A : Type u_6
inst✝¹ : CommRing A
inst✝ : Algebra R A
h : Function.Surjective ⇑(algebraMap R A)
g : A →ₐ[R] R ⧸ RingHom.ker (ofId R A).toRingHom ^ 2
hg : (ofId R A).kerSquareLift.comp g = AlgHom.id R A
⊢ (g.comp (ofId R A).kerSquareLift).comp (Ideal.Quotient.mkₐ R (RingHom.ker (algebraMap R A) ^ 2)) =
    (AlgHom.id R (R ⧸ RingHom.ker (algebraMap R A) ^ 2)).comp (Ideal.Quotient.mkₐ R (RingHom.ker (algebraMap R A) ^ 2))

Tactic:
by

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

Goal state before:
pp: R : Type u_4
inst✝² : CommRing R
A : Type u_6
inst✝¹ : CommRing A
inst✝ : Algebra R A
h : Function.Surjective ⇑(algebraMap R A)
g : A →ₐ[R] R ⧸ RingHom.ker (ofId R A).toRingHom ^ 2
hg : (ofId R A).kerSquareLift.comp g = AlgHom.id R A
e : A ≃ₐ[R] R ⧸ RingHom.ker (algebraMap R A) ^ 2 := AlgEquiv.ofAlgHom g (ofId R A).kerSquareLift ⋯ hg
⊢ Function.Injective ⇑↑↑e

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Kaehler.JacobiZariski

Goal state before:
pp: R : Type u₁
S : Type u₂
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
T : Type u₃
inst✝³ : CommRing T
inst✝² : Algebra R T
inst✝¹ : Algebra S T
inst✝ : IsScalarTower R S T
ι : Type w₁
σ : Type w₂
Q : Generators S T ι
P : Generators R S σ
⊢ LinearMap.fst T Q.toExtension.CotangentSpace (T ⊗[S] P.toExtension.CotangentSpace) ∘ₗ ↑(compEquiv Q P) =
    Extension.CotangentSpace.map (Q.ofComp P).toExtensionHom

Tactic:
by

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

Goal state before:
pp: R : Type u_4
inst✝² : CommRing R
A : Type u_6
inst✝¹ : CommRing A
inst✝ : Algebra R A
h : Function.Surjective ⇑(algebraMap R A)
H : IsIdempotentElem (RingHom.ker (algebraMap R A))
e : (R ⧸ RingHom.ker (algebraMap R A) ^ 2) ≃ₐ[R] A :=
  (Ideal.quotientEquivAlgOfEq R ⋯).trans (Ideal.quotientKerAlgEquivOfSurjective h)
x : R
⊢ ((ofId R A).kerSquareLift.comp ↑e.symm) ((algebraMap R A) x) = (AlgHom.id R A) ((algebraMap R A) x)

Tactic:
by

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

Goal state before:
pp: R : Type u_4
inst✝² : CommRing R
A : Type u_6
inst✝¹ : CommRing A
inst✝ : Algebra R A
h : Function.Surjective ⇑(algebraMap R A)
⊢ FormallySmooth R A ↔ IsIdempotentElem (RingHom.ker (algebraMap R A))

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Kaehler.JacobiZariski

Goal state before:
pp: R : Type u₁
S : Type u₂
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
T : Type u₃
inst✝³ : CommRing T
inst✝² : Algebra R T
inst✝¹ : Algebra S T
inst✝ : IsScalarTower R S T
ι : Type w₁
σ : Type w₂
Q : Generators S T ι
P : Generators R S σ
⊢ Function.Injective ⇑(LinearMap.liftBaseChange T (Extension.CotangentSpace.map (Q.toComp P).toExtensionHom))

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Kaehler.JacobiZariski

Goal state before:
pp: R : Type u₁
S : Type u₂
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
T : Type u₃
inst✝³ : CommRing T
inst✝² : Algebra R T
inst✝¹ : Algebra S T
inst✝ : IsScalarTower R S T
ι : Type w₁
σ : Type w₂
Q : Generators S T ι
P : Generators R S σ
⊢ Function.Surjective ⇑(Extension.CotangentSpace.map (Q.ofComp P).toExtensionHom)

Tactic:
by

Goal state after: