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

Goal state before:
pp: R : Type u_1
P : Type u_2
S : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing P
inst✝⁴ : CommRing S
inst✝³ : Algebra R P
inst✝² : Algebra P S
inst✝¹ : Algebra R S
inst✝ : IsScalarTower R P S
g✝ : S →ₐ[R] P
f : P →ₐ[R] S
hf' : RingHom.ker f ^ 2 = ⊥
g : S →ₐ[R] P
hg : f.comp g = AlgHom.id R S
x : R
y : P
⊢ ⟨x • y - g (f (x • y)), ⋯⟩ = (RingHom.id R) x • ⟨y - g (f y), ⋯⟩

Tactic:
ext

Goal state after:
R : Type u_1
P : Type u_2
S : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing P
inst✝⁴ : CommRing S
inst✝³ : Algebra R P
inst✝² : Algebra P S
inst✝¹ : Algebra R S
inst✝ : IsScalarTower R P S
g✝ : S →ₐ[R] P
f : P →ₐ[R] S
hf' : RingHom.ker f ^ 2 = ⊥
g : S →ₐ[R] P
hg : f.comp g = AlgHom.id R S
x : R
y : P
⊢ ↑⟨x • y - g (f (x • y)), ⋯⟩ = ↑((RingHom.id R) x • ⟨y - g (f y), ⋯⟩)
Module: Mathlib.RingTheory.Smooth.Kaehler

Goal state before:
pp: R : Type u_1
P : Type u_2
S : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing P
inst✝⁴ : CommRing S
inst✝³ : Algebra R P
inst✝² : Algebra P S
inst✝¹ : Algebra R S
inst✝ : IsScalarTower R P S
g✝ : S →ₐ[R] P
f : P →ₐ[R] S
hf' : RingHom.ker f ^ 2 = ⊥
g : S →ₐ[R] P
hg : f.comp g = AlgHom.id R S
a b : P
this : (a - g (f a)) * (b - g (f b)) = 0
⊢ { toFun := fun x ↦ ⟨x - g (f x), ⋯⟩, map_add' := ⋯, map_smul' := ⋯ } (a * b) =
    a • { toFun := fun x ↦ ⟨x - g (f x), ⋯⟩, map_add' := ⋯, map_smul' := ⋯ } b +
      b • { toFun := fun x ↦ ⟨x - g (f x), ⋯⟩, map_add' := ⋯, map_smul' := ⋯ } a

Tactic:
ext

Goal state after:
R : Type u_1
P : Type u_2
S : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing P
inst✝⁴ : CommRing S
inst✝³ : Algebra R P
inst✝² : Algebra P S
inst✝¹ : Algebra R S
inst✝ : IsScalarTower R P S
g✝ : S →ₐ[R] P
f : P →ₐ[R] S
hf' : RingHom.ker f ^ 2 = ⊥
g : S →ₐ[R] P
hg : f.comp g = AlgHom.id R S
a b : P
this : (a - g (f a)) * (b - g (f b)) = 0
⊢ ↑({ toFun := fun x ↦ ⟨x - g (f x), ⋯⟩, map_add' := ⋯, map_smul' := ⋯ } (a * b)) =
    ↑(a • { toFun := fun x ↦ ⟨x - g (f x), ⋯⟩, map_add' := ⋯, map_smul' := ⋯ } b +
        b • { toFun := fun x ↦ ⟨x - g (f x), ⋯⟩, map_add' := ⋯, map_smul' := ⋯ } a)
Module: Mathlib.RingTheory.Smooth.Kaehler

Goal state before:
pp: R : Type u_1
P : Type u_2
S : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing P
inst✝⁴ : CommRing S
inst✝³ : Algebra R P
inst✝² : Algebra P S
inst✝¹ : Algebra R S
inst✝ : IsScalarTower R P S
g : S →ₐ[R] P
hf' : RingHom.ker (algebraMap P S) ^ 2 = ⊥
hg : (IsScalarTower.toAlgHom R P S).comp g = AlgHom.id R S
this : Algebra S P := g.toAlgebra
⊢ IsScalarTower P S ↥(RingHom.ker (algebraMap P S))

Tactic:
constructor

Goal state after:
R : Type u_1
P : Type u_2
S : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing P
inst✝⁴ : CommRing S
inst✝³ : Algebra R P
inst✝² : Algebra P S
inst✝¹ : Algebra R S
inst✝ : IsScalarTower R P S
g : S →ₐ[R] P
hf' : RingHom.ker (algebraMap P S) ^ 2 = ⊥
hg : (IsScalarTower.toAlgHom R P S).comp g = AlgHom.id R S
this : Algebra S P := ⋯
⊢ ∀ (x : P) (y : S) (z : ↥(RingHom.ker (algebraMap P S))), (x • y) • z = x • y • z
Module: Mathlib.RingTheory.Smooth.Kaehler

Goal state before:
pp: R : Type u_1
P : Type u_2
S : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing P
inst✝⁴ : CommRing S
inst✝³ : Algebra R P
inst✝² : Algebra P S
inst✝¹ : Algebra R S
inst✝ : IsScalarTower R P S
g : S →ₐ[R] P
hf' : RingHom.ker (algebraMap P S) ^ 2 = ⊥
hg : (IsScalarTower.toAlgHom R P S).comp g = AlgHom.id R S
this : Algebra S P := g.toAlgebra
p : P
s : S
m : ↥(RingHom.ker (algebraMap P S))
⊢ (p • s) • m = p • s • m

Tactic:
ext

Goal state after:
R : Type u_1
P : Type u_2
S : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing P
inst✝⁴ : CommRing S
inst✝³ : Algebra R P
inst✝² : Algebra P S
inst✝¹ : Algebra R S
inst✝ : IsScalarTower R P S
g : S →ₐ[R] P
hf' : RingHom.ker (algebraMap P S) ^ 2 = ⊥
hg : (IsScalarTower.toAlgHom R P S).comp g = AlgHom.id R S
this : Algebra S P := ⋯
p : P
s : S
m : ↥(RingHom.ker (algebraMap P S))
⊢ ↑((p • s) • m) = ↑(p • s • m)
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
ι' : Type w₃
σ : Type w₂
σ' : Type w₄
τ : Type w₅
Q : Generators S T ι
P : Generators R S σ
Q' : Generators S T ι'
P' : Generators R S σ'
W : Generators R T τ
⊢ Extension.CotangentSpace.map (Q.ofComp P).toExtensionHom ∘ₗ (Q.comp P).toExtension.cotangentComplex =
    Q.toExtension.cotangentComplex ∘ₗ Extension.Cotangent.map (Q.ofComp P).toExtensionHom

Tactic:
ext

Goal state after:
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
ι' : Type w₃
σ : Type w₂
σ' : Type w₄
τ : Type w₅
Q : Generators S T ι
P : Generators R S σ
Q' : Generators S T ι'
P' : Generators R S σ'
W : Generators R T τ
x✝ : (Q.comp P).toExtension.Cotangent
⊢ (Extension.CotangentSpace.map (Q.ofComp P).toExtensionHom ∘ₗ (Q.comp P).toExtension.cotangentComplex) x✝ =
    (Q.toExtension.cotangentComplex ∘ₗ Extension.Cotangent.map (Q.ofComp P).toExtensionHom) x✝
Module: Mathlib.RingTheory.Unramified.Field

Goal state before:
pp: K : Type u_1
A : Type u_2
inst✝⁶ : Field K
inst✝⁵ : CommRing A
inst✝⁴ : Algebra K A
inst✝³ : FormallyUnramified K A
inst✝² : EssFiniteType K A
inst✝¹ : IsAlgClosed K
inst✝ : IsLocalRing A
this✝ : Module.Finite K A
this : IsArtinianRing A
hA : IsNilpotent (IsLocalRing.maximalIdeal A)
e : K ≃ₐ[K] A ⧸ IsLocalRing.maximalIdeal A :=
  let __spread.0 := ofId K (A ⧸ IsLocalRing.maximalIdeal A);
  let __spread.1 := Equiv.ofBijective ⇑(algebraMap K (A ⧸ IsLocalRing.maximalIdeal A)) ⋯;
  { toFun := (↑↑__spread.0.toRingHom).toFun, invFun := __spread.1.invFun, left_inv := ⋯, right_inv := ⋯, map_mul' := ⋯,
    map_add' := ⋯, commutes' := ⋯ }
e' : A ⊗[K] (A ⧸ IsLocalRing.maximalIdeal A) ≃ₐ[A] A :=
  (TensorProduct.congr AlgEquiv.refl e.symm).trans (TensorProduct.rid K A A)
f : A ⧸ IsLocalRing.maximalIdeal A →ₗ[A] A := e'.toLinearMap ∘ₗ sec K A (A ⧸ IsLocalRing.maximalIdeal A)
⊢ (ofId A (A ⧸ IsLocalRing.maximalIdeal A)).toLinearMap ∘ₗ e'.toLinearMap ∘ₗ sec K A (A ⧸ IsLocalRing.maximalIdeal A) =
    LinearMap.id

Tactic:
rw [← LinearMap.comp_assoc, ← comp_sec K A]

Goal state after:
K : Type u_1
A : Type u_2
inst✝⁶ : Field K
inst✝⁵ : CommRing A
inst✝⁴ : Algebra K A
inst✝³ : FormallyUnramified K A
inst✝² : EssFiniteType K A
inst✝¹ : IsAlgClosed K
inst✝ : IsLocalRing A
this✝ : Module.Finite K A
this : IsArtinianRing A
hA : IsNilpotent (IsLocalRing.maximalIdeal A)
e : K ≃ₐ[K] A ⧸ IsLocalRing.maximalIdeal A :=
  let __spread.0 := ofId K (A ⧸ IsLocalRing.maximalIdeal A);
  let __spread.1 := Equiv.ofBijective ⇑(algebraMap K (A ⧸ IsLocalRing.maximalIdeal A)) ⋯;
  { toFun := (↑↑__spread.0.toRingHom).toFun, invFun := __spread.1.invFun, left_inv := ⋯, right_inv := ⋯, map_mul' := ⋯,
    map_add' := ⋯, commutes' := ⋯ }
e' : A ⊗[K] (A ⧸ IsLocalRing.maximalIdeal A) ≃ₐ[A] A :=
  (TensorProduct.congr AlgEquiv.refl e.symm).trans (TensorProduct.rid K A A)
f : A ⧸ IsLocalRing.maximalIdeal A →ₗ[A] A := e'.toLinearMap ∘ₗ sec K A (A ⧸ IsLocalRing.maximalIdeal A)
⊢ ((ofId A (A ⧸ IsLocalRing.maximalIdeal A)).toLinearMap ∘ₗ e'.toLinearMap) ∘ₗ
      sec K A (A ⧸ IsLocalRing.maximalIdeal A) =
    TensorProduct.AlgebraTensorModule.lift (↑K (lsmul A A (A ⧸ IsLocalRing.maximalIdeal A)).toLinearMap.flip).flip ∘ₗ
      sec K A (A ⧸ IsLocalRing.maximalIdeal A)
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
i : σ
| (Extension.CotangentSpace.map (Q.toComp P).toExtensionHom) (1 ⊗ₜ[P.Ring] (D R P.Ring) (X i))

Tactic:
tactic => exact Extension.CotangentSpace.map_tmul ..

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

Goal state before:
pp: case hz
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.ker
hx : Extension.Cotangent.mk x ∈ Q.toExtension.cotangentComplex.ker
y : (Q.comp P).toExtension.Cotangent := Extension.Cotangent.mk (Q.kerCompPreimage P x)
hy : (Extension.Cotangent.map (Q.ofComp P).toExtensionHom) y = Extension.Cotangent.mk x
z : T ⊗[S] P.toExtension.CotangentSpace :=
  ((CotangentSpace.compEquiv Q P) ((Q.comp P).toExtension.cotangentComplex y)).2
⊢ (0, ((CotangentSpace.compEquiv Q P) ((Q.comp P).toExtension.cotangentComplex y)).2) =
    (CotangentSpace.compEquiv Q P) ((Q.comp P).toExtension.cotangentComplex y)

Tactic:
ext

Goal state after:
case hz.fst
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.ker
hx : Extension.Cotangent.mk x ∈ Q.toExtension.cotangentComplex.ker
y : (Q.comp P).toExtension.Cotangent := ⋯
hy : (Extension.Cotangent.map (Q.ofComp P).toExtensionHom) y = Extension.Cotangent.mk x
z : T ⊗[S] P.toExtension.CotangentSpace := ⋯
⊢ (0, ((CotangentSpace.compEquiv Q P) ((Q.comp P).toExtension.cotangentComplex y)).2).1 =
    ((CotangentSpace.compEquiv Q P) ((Q.comp P).toExtension.cotangentComplex y)).1

case hz.snd
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.ker
hx : Extension.Cotangent.mk x ∈ Q.toExtension.cotangentComplex.ker
y : (Q.comp P).toExtension.Cotangent := ⋯
hy : (Extension.Cotangent.map (Q.ofComp P).toExtensionHom) y = Extension.Cotangent.mk x
z : T ⊗[S] P.toExtension.CotangentSpace := ⋯
⊢ (0, ((CotangentSpace.compEquiv Q P) ((Q.comp P).toExtension.cotangentComplex y)).2).2 =
    ((CotangentSpace.compEquiv Q P) ((Q.comp P).toExtension.cotangentComplex y)).2
Module: Mathlib.RingTheory.Unramified.Field

Goal state before:
pp: K : Type u_1
A : Type u_2
inst✝⁴ : Field K
inst✝³ : CommRing A
inst✝² : Algebra K A
inst✝¹ : FormallyUnramified K A
inst✝ : EssFiniteType K A
⊢ IsReduced A

Tactic:
constructor

Goal state after:
K : Type u_1
A : Type u_2
inst✝⁴ : Field K
inst✝³ : CommRing A
inst✝² : Algebra K A
inst✝¹ : FormallyUnramified K A
inst✝ : EssFiniteType K A
⊢ ∀ (x : A), IsNilpotent x → x = 0
Module: Mathlib.RingTheory.Etale.Kaehler

Goal state before:
pp: R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing T
inst✝⁴ : Algebra R S
inst✝³ : Algebra R T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R S T
inst✝ : Algebra.FormallyEtale S T
⊢ lift (↑S ((Algebra.linearMap T (Module.End T (Ω[S⁄R] →ₗ[S] Ω[T⁄R]))).flip (map R R S T))) =
    ↑S ↑(tensorKaehlerEquivOfFormallyEtale R S T)

Tactic:
ext

Goal state after:
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing T
inst✝⁴ : Algebra R S
inst✝³ : Algebra R T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R S T
inst✝ : Algebra.FormallyEtale S T
x✝ : T
a✝ : S
⊢ (((AlgebraTensorModule.curry
              (lift (↑S ((Algebra.linearMap T (Module.End T (Ω[S⁄R] →ₗ[S] Ω[T⁄R]))).flip (map R R S T)))))
            x✝).compDer
        (D R S))
      a✝ =
    (((AlgebraTensorModule.curry (↑S ↑(tensorKaehlerEquivOfFormallyEtale R S T))) x✝).compDer (D R S)) a✝
Module: Mathlib.RingTheory.Unramified.Field

Goal state before:
pp: K : Type u_1
L : Type u_3
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : FormallyUnramified K L
inst✝ : EssFiniteType K L
this✝¹ : Module.Finite K L
this✝ : FormallyUnramified (↥(separableClosure K L)) L
this : EssFiniteType (↥(separableClosure K L)) L
⊢ separableClosure K L = ⊤

Tactic:
ext

Goal state after:
K : Type u_1
L : Type u_3
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : FormallyUnramified K L
inst✝ : EssFiniteType K L
this✝¹ : Module.Finite K L
this✝ : FormallyUnramified (↥(separableClosure K L)) L
this : EssFiniteType (↥(separableClosure K L)) L
x✝ : L
⊢ x✝ ∈ separableClosure K L ↔ x✝ ∈ ⊤
Module: Mathlib.RingTheory.Etale.Kaehler

Goal state before:
pp: R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : CommRing T
inst✝³ : Algebra R S
inst✝² : Algebra R T
inst✝¹ : Algebra S T
inst✝ : IsScalarTower R S T
P : Extension R S
Q : Extension R T
f : P.Hom Q
alg : Algebra P.Ring Q.Ring
halg : algebraMap P.Ring Q.Ring = f.toRingHom
H : Function.Bijective ⇑(LinearMap.liftBaseChange Q.Ring (f.mapKer halg))
this✝¹ : Algebra P.Ring T := ((algebraMap S T).comp (algebraMap P.Ring S)).toAlgebra
this✝ : IsScalarTower P.Ring S T
this : IsScalarTower P.Ring Q.Ring T
e : Q.Ring ⊗[P.Ring] ↥P.ker ≃ₗ[Q.Ring] ↥Q.ker :=
  LinearEquiv.ofBijective (LinearMap.liftBaseChange Q.Ring (f.mapKer halg)) H
f' : ↥Q.ker →ₗ[Q.Ring] T ⊗[S] P.Cotangent :=
  LinearMap.liftBaseChange Q.Ring (↑P.Ring ((TensorProduct.mk S T P.Cotangent) 1) ∘ₗ Cotangent.mk) ∘ₗ ↑e.symm
a : Q.Ring
b : ↥P.ker
c : Q.Ring
d : ↥P.ker
⊢ ↑b • Cotangent.mk d = 0

Tactic:
ext

Goal state after:
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : CommRing T
inst✝³ : Algebra R S
inst✝² : Algebra R T
inst✝¹ : Algebra S T
inst✝ : IsScalarTower R S T
P : Extension R S
Q : Extension R T
f : P.Hom Q
alg : Algebra P.Ring Q.Ring
halg : algebraMap P.Ring Q.Ring = f.toRingHom
H : Function.Bijective ⇑(LinearMap.liftBaseChange Q.Ring (f.mapKer halg))
this✝¹ : Algebra P.Ring T := ⋯
this✝ : IsScalarTower P.Ring S T
this : IsScalarTower P.Ring Q.Ring T
e : Q.Ring ⊗[P.Ring] ↥P.ker ≃ₗ[Q.Ring] ↥Q.ker := ⋯
f' : ↥Q.ker →ₗ[Q.Ring] T ⊗[S] P.Cotangent := ⋯
a : Q.Ring
b : ↥P.ker
c : Q.Ring
d : ↥P.ker
⊢ (↑b • Cotangent.mk d).val = Cotangent.val 0
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 : P.Ring
x_in : x ∈ P.toExtension.ker
property✝ : Extension.Cotangent.mk ⟨x, x_in⟩ ∈ P.toExtension.cotangentComplex.ker
⊢ (Extension.H1Cotangent.map (Q.toComp P).toExtensionHom) ⟨Extension.Cotangent.mk ⟨x, x_in⟩, property✝⟩ ∈
    ↑(Extension.H1Cotangent.map (Q.ofComp P).toExtensionHom).ker

Tactic:
ext

Goal state after:
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 : P.Ring
x_in : x ∈ P.toExtension.ker
property✝ : Extension.Cotangent.mk ⟨x, x_in⟩ ∈ P.toExtension.cotangentComplex.ker
⊢ (↑((Extension.H1Cotangent.map (Q.ofComp P).toExtensionHom)
          ((Extension.H1Cotangent.map (Q.toComp P).toExtensionHom)
            ⟨Extension.Cotangent.mk ⟨x, x_in⟩, property✝⟩))).val =
    (↑0).val
Module: Mathlib.RingTheory.Smooth.Kaehler

Goal state before:
pp: R : Type u_1
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
P : Extension R S
⊢ Function.Bijective ⇑(map P.toInfinitesimal)

Tactic:
constructor

Goal state after:
case left
R : Type u_1
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
P : Extension R S
⊢ Function.Injective ⇑(map P.toInfinitesimal)

case right
R : Type u_1
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
P : Extension R S
⊢ Surjective ⇑(map P.toInfinitesimal)
Module: Mathlib.RingTheory.Smooth.Kaehler

Goal state before:
pp: case left
R : Type u_1
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
P : Extension R S
x : ↥P.ker
hx✝ : (map P.toInfinitesimal) (mk x) = 0
hx : ↑x ∈ P.ker ^ 2
⊢ mk x = 0

Tactic:
ext

Goal state after:
case left
R : Type u_1
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
P : Extension R S
x : ↥P.ker
hx✝ : (map P.toInfinitesimal) (mk x) = 0
hx : ↑x ∈ P.ker ^ 2
⊢ (mk x).val = val 0
Module: Mathlib.RingTheory.Unramified.LocalRing

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 : IsLocalRing R
inst✝³ : IsLocalRing S
inst✝² : IsLocalHom (algebraMap R S)
inst✝¹ : EssFiniteType R S
inst✝ : Algebra.IsSeparable (ResidueField R) (ResidueField S)
H : Ideal.map (algebraMap R S) (maximalIdeal R) = maximalIdeal S
⊢ FormallyUnramified R S

Tactic:
constructor

Goal state after:
R : Type u_1
S : Type u_2
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : IsLocalRing R
inst✝³ : IsLocalRing S
inst✝² : IsLocalHom (algebraMap R S)
inst✝¹ : EssFiniteType R S
inst✝ : Algebra.IsSeparable (ResidueField R) (ResidueField S)
H : Ideal.map (algebraMap R S) (maximalIdeal R) = maximalIdeal S
⊢ Subsingleton Ω[S⁄R]
Module: Mathlib.RingTheory.Smooth.Kaehler

Goal state before:
pp: case right
R : Type u_1
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
P : Extension R S
x : P.infinitesimal.Cotangent
⊢ ∃ a, (map P.toInfinitesimal) a = x

Tactic:
obtain ⟨⟨x, hx⟩, rfl⟩ := Cotangent.mk_surjective x

Goal state after:
case right
R : Type u_1
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
P : Extension R S
x : P.infinitesimal.Ring
hx : x ∈ P.infinitesimal.ker
⊢ ∃ a, (map P.toInfinitesimal) a = mk ⟨x, hx⟩
Module: Mathlib.RingTheory.Kaehler.JacobiZariski

Goal state before:
pp: case h.zero
R : Type u₁
S : Type u₂
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
T : Type u₃
inst✝⁴ : CommRing T
inst✝³ : Algebra R T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R S T
ι : Type w₁
σ : Type w₂
Q : Generators S T ι
P : Generators R S σ
inst✝ : Flat S T
x_in :
  (liftBaseChange T (Extension.Cotangent.map (Q.toComp P).toExtensionHom)) ((lTensor T Extension.h1Cotangentι) 0) ∈
    (Q.comp P).toExtension.cotangentComplex.ker
⊢ (liftBaseChange T (Extension.H1Cotangent.map (Q.toComp P).toExtensionHom)) 0 =
    ⟨(liftBaseChange T (Extension.Cotangent.map (Q.toComp P).toExtensionHom)) ((lTensor T Extension.h1Cotangentι) 0),
      x_in⟩

Tactic:
ext

Goal state after:
case h.zero
R : Type u₁
S : Type u₂
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
T : Type u₃
inst✝⁴ : CommRing T
inst✝³ : Algebra R T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R S T
ι : Type w₁
σ : Type w₂
Q : Generators S T ι
P : Generators R S σ
inst✝ : Flat S T
x_in :
  (liftBaseChange T (Extension.Cotangent.map (Q.toComp P).toExtensionHom)) ((lTensor T Extension.h1Cotangentι) 0) ∈
    (Q.comp P).toExtension.cotangentComplex.ker
⊢ (↑((liftBaseChange T (Extension.H1Cotangent.map (Q.toComp P).toExtensionHom)) 0)).val =
    (↑⟨(liftBaseChange T (Extension.Cotangent.map (Q.toComp P).toExtensionHom)) ((lTensor T Extension.h1Cotangentι) 0),
          x_in⟩).val
Module: Mathlib.RingTheory.Etale.Kaehler

Goal state before:
pp: R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing T
inst✝⁴ : Algebra R S
inst✝³ : Algebra R T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R S T
P : Extension R S
Q : Extension R T
f : P.Hom Q
alg : Algebra P.Ring Q.Ring
halg : algebraMap P.Ring Q.Ring = f.toRingHom
inst✝ : Module.Flat S T
H₁ : f.toRingHom.FormallyEtale
H₂ : Function.Bijective ⇑(LinearMap.liftBaseChange Q.Ring (f.mapKer halg))
⊢ Function.Bijective ⇑(LinearMap.liftBaseChange T (H1Cotangent.map f))

Tactic:
constructor

Goal state after:
case left
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing T
inst✝⁴ : Algebra R S
inst✝³ : Algebra R T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R S T
P : Extension R S
Q : Extension R T
f : P.Hom Q
alg : Algebra P.Ring Q.Ring
halg : algebraMap P.Ring Q.Ring = f.toRingHom
inst✝ : Module.Flat S T
H₁ : f.toRingHom.FormallyEtale
H₂ : Function.Bijective ⇑(LinearMap.liftBaseChange Q.Ring (f.mapKer halg))
⊢ Function.Injective ⇑(LinearMap.liftBaseChange T (H1Cotangent.map f))

case right
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing T
inst✝⁴ : Algebra R S
inst✝³ : Algebra R T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R S T
P : Extension R S
Q : Extension R T
f : P.Hom Q
alg : Algebra P.Ring Q.Ring
halg : algebraMap P.Ring Q.Ring = f.toRingHom
inst✝ : Module.Flat S T
H₁ : f.toRingHom.FormallyEtale
H₂ : Function.Bijective ⇑(LinearMap.liftBaseChange Q.Ring (f.mapKer halg))
⊢ Function.Surjective ⇑(LinearMap.liftBaseChange T (H1Cotangent.map f))
Module: Mathlib.RingTheory.Kaehler.JacobiZariski

Goal state before:
pp: case h.tmul
R : Type u₁
S : Type u₂
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
T : Type u₃
inst✝⁴ : CommRing T
inst✝³ : Algebra R T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R S T
ι : Type w₁
σ : Type w₂
Q : Generators S T ι
P : Generators R S σ
inst✝ : Flat S T
x : T
y : P.toExtension.H1Cotangent
x_in :
  (liftBaseChange T (Extension.Cotangent.map (Q.toComp P).toExtensionHom))
      ((lTensor T Extension.h1Cotangentι) (x ⊗ₜ[S] y)) ∈
    (Q.comp P).toExtension.cotangentComplex.ker
⊢ (liftBaseChange T (Extension.H1Cotangent.map (Q.toComp P).toExtensionHom)) (x ⊗ₜ[S] y) =
    ⟨(liftBaseChange T (Extension.Cotangent.map (Q.toComp P).toExtensionHom))
        ((lTensor T Extension.h1Cotangentι) (x ⊗ₜ[S] y)),
      x_in⟩

Tactic:
ext

Goal state after:
case h.tmul
R : Type u₁
S : Type u₂
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
T : Type u₃
inst✝⁴ : CommRing T
inst✝³ : Algebra R T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R S T
ι : Type w₁
σ : Type w₂
Q : Generators S T ι
P : Generators R S σ
inst✝ : Flat S T
x : T
y : P.toExtension.H1Cotangent
x_in :
  (liftBaseChange T (Extension.Cotangent.map (Q.toComp P).toExtensionHom))
      ((lTensor T Extension.h1Cotangentι) (x ⊗ₜ[S] y)) ∈
    (Q.comp P).toExtension.cotangentComplex.ker
⊢ (↑((liftBaseChange T (Extension.H1Cotangent.map (Q.toComp P).toExtensionHom)) (x ⊗ₜ[S] y))).val =
    (↑⟨(liftBaseChange T (Extension.Cotangent.map (Q.toComp P).toExtensionHom))
            ((lTensor T Extension.h1Cotangentι) (x ⊗ₜ[S] y)),
          x_in⟩).val
Module: Mathlib.RingTheory.Etale.Kaehler

Goal state before:
pp: case right
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing T
inst✝⁴ : Algebra R S
inst✝³ : Algebra R T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R S T
P : Extension R S
Q : Extension R T
f : P.Hom Q
alg : Algebra P.Ring Q.Ring
halg : algebraMap P.Ring Q.Ring = f.toRingHom
inst✝ : Module.Flat S T
H₁ : f.toRingHom.FormallyEtale
H₂ : Function.Bijective ⇑(LinearMap.liftBaseChange Q.Ring (f.mapKer halg))
x : Q.H1Cotangent
this : Function.Exact ⇑(LinearMap.baseChange T h1Cotangentι) ⇑(LinearMap.baseChange T P.cotangentComplex)
a : T ⊗[S] P.H1Cotangent
ha : (LinearMap.baseChange T h1Cotangentι) a = (tensorCotangent f halg H₂).symm ↑x
⊢ ↑((LinearMap.liftBaseChange T (H1Cotangent.map f)) a) =
    (tensorCotangent f halg H₂) ((LinearMap.baseChange T h1Cotangentι) a)

Tactic:
change (h1Cotangentι ∘ₗ (H1Cotangent.map f).liftBaseChange T) _ =
      ((Cotangent.map f).liftBaseChange T ∘ₗ h1Cotangentι.baseChange T) _

Goal state after:
case right
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing T
inst✝⁴ : Algebra R S
inst✝³ : Algebra R T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R S T
P : Extension R S
Q : Extension R T
f : P.Hom Q
alg : Algebra P.Ring Q.Ring
halg : algebraMap P.Ring Q.Ring = f.toRingHom
inst✝ : Module.Flat S T
H₁ : f.toRingHom.FormallyEtale
H₂ : Function.Bijective ⇑(LinearMap.liftBaseChange Q.Ring (f.mapKer halg))
x : Q.H1Cotangent
this : Function.Exact ⇑(LinearMap.baseChange T h1Cotangentι) ⇑(LinearMap.baseChange T P.cotangentComplex)
a : T ⊗[S] P.H1Cotangent
ha : (LinearMap.baseChange T h1Cotangentι) a = (tensorCotangent f halg H₂).symm ↑x
⊢ (h1Cotangentι ∘ₗ LinearMap.liftBaseChange T (H1Cotangent.map f)) a =
    (LinearMap.liftBaseChange T (Cotangent.map f) ∘ₗ LinearMap.baseChange T h1Cotangentι) a