Module: Mathlib.AlgebraicGeometry.Normalization

Goal state before:
pp: case h₀
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
inst✝⁴ : QuasiCompact f✝
inst✝³ : QuasiSeparated f✝
X S Y : Scheme
f : X ⟶ S
g : Y ⟶ S
inst✝² : QuasiCompact f
inst✝¹ : QuasiSeparated f
inst✝ : Smooth g
x : ↥(pullback (fromNormalization f) g)
U : TopologicalSpace.Opens ↥S
hU : U ∈ S.affineOpens
hxU : (pullback.snd (fromNormalization f) g ≫ g) x ∈ ↑U
V : TopologicalSpace.Opens ↥Y
hV : V ∈ Y.affineOpens
hxV : (pullback.snd (fromNormalization f) g) x ∈ ↑V
hVU : V ≤ g ⁻¹ᵁ U
W : (pullback (fromNormalization f) g).Opens := ⋯
this✝¹ : IsIso (pushoutSection ⋯ hVU ⋯ ⋯)
this✝ : (CommRingCat.Hom.hom (appLE g U V hVU)).Smooth
algInst✝² : Algebra ↑Γ(S, U) ↑Γ(X, f ⁻¹ᵁ U) := ⋯
algInst✝¹ : Algebra ↑Γ(S, U) ↑Γ(Y, V) := ⋯
algInst✝ : Algebra ↑Γ(Y, V) ↑Γ(pullback f g, pullback.snd f g ⁻¹ᵁ V) := ⋯
algebraizeInst✝ : Algebra.Smooth ↑Γ(S, U) ↑Γ(Y, V)
this : IsIso (pushoutSection ⋯ hVU ⋯ ⋯)
e₀ : CommRingCat.of
    (↑(Y.presheaf.1 (Opposite.op V)) ⊗[↑(S.presheaf.1 (Opposite.op U))] ↑(X.presheaf.1 (Opposite.op (f ⁻¹ᵁ U)))) ≅
  Γ(pullback f g, pullback.snd f g ⁻¹ᵁ V) := ⋯
e : ↑Γ(Y, V) ⊗[↑Γ(S, U)] ↑Γ(X, f ⁻¹ᵁ U) ≃ₐ[↑Γ(Y, V)] ↑Γ(pullback f g, pullback.snd f g ⁻¹ᵁ V) := ⋯
ψ : ↑Γ(Y, V) ⊗[↑Γ(S, U)] ↥(integralClosure ↑Γ(S, U) ↑Γ(X, f ⁻¹ᵁ U)) →ₐ[↑Γ(Y, V)]
  ↥(integralClosure ↑Γ(Y, V) ↑Γ(pullback f g, pullback.snd f g ⁻¹ᵁ V)) := ⋯
hψ : Function.Bijective ⇑ψ
φ : pushout (app (fromNormalization f) U) (appLE g U V hVU) ⟶
  Γ(normalization (pullback.snd f g), normalizationPullback f g ⁻¹ᵁ W) := ⋯
⊢ pushout.inl (app (fromNormalization f) U) (appLE g U V hVU) ≫
      (pushout.congrHom ⋯ ⋯).hom ≫ pushoutSection ⋯ hVU ⋯ ⋯ ≫ app (normalizationPullback f g) W =
    pushout.inl (app (fromNormalization f) U) (appLE g U V hVU) ≫ φ

pp: case h₁
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
inst✝⁴ : QuasiCompact f✝
inst✝³ : QuasiSeparated f✝
X S Y : Scheme
f : X ⟶ S
g : Y ⟶ S
inst✝² : QuasiCompact f
inst✝¹ : QuasiSeparated f
inst✝ : Smooth g
x : ↥(pullback (fromNormalization f) g)
U : TopologicalSpace.Opens ↥S
hU : U ∈ S.affineOpens
hxU : (pullback.snd (fromNormalization f) g ≫ g) x ∈ ↑U
V : TopologicalSpace.Opens ↥Y
hV : V ∈ Y.affineOpens
hxV : (pullback.snd (fromNormalization f) g) x ∈ ↑V
hVU : V ≤ g ⁻¹ᵁ U
W : (pullback (fromNormalization f) g).Opens := ⋯
this✝¹ : IsIso (pushoutSection ⋯ hVU ⋯ ⋯)
this✝ : (CommRingCat.Hom.hom (appLE g U V hVU)).Smooth
algInst✝² : Algebra ↑Γ(S, U) ↑Γ(X, f ⁻¹ᵁ U) := ⋯
algInst✝¹ : Algebra ↑Γ(S, U) ↑Γ(Y, V) := ⋯
algInst✝ : Algebra ↑Γ(Y, V) ↑Γ(pullback f g, pullback.snd f g ⁻¹ᵁ V) := ⋯
algebraizeInst✝ : Algebra.Smooth ↑Γ(S, U) ↑Γ(Y, V)
this : IsIso (pushoutSection ⋯ hVU ⋯ ⋯)
e₀ : CommRingCat.of
    (↑(Y.presheaf.1 (Opposite.op V)) ⊗[↑(S.presheaf.1 (Opposite.op U))] ↑(X.presheaf.1 (Opposite.op (f ⁻¹ᵁ U)))) ≅
  Γ(pullback f g, pullback.snd f g ⁻¹ᵁ V) := ⋯
e : ↑Γ(Y, V) ⊗[↑Γ(S, U)] ↑Γ(X, f ⁻¹ᵁ U) ≃ₐ[↑Γ(Y, V)] ↑Γ(pullback f g, pullback.snd f g ⁻¹ᵁ V) := ⋯
ψ : ↑Γ(Y, V) ⊗[↑Γ(S, U)] ↥(integralClosure ↑Γ(S, U) ↑Γ(X, f ⁻¹ᵁ U)) →ₐ[↑Γ(Y, V)]
  ↥(integralClosure ↑Γ(Y, V) ↑Γ(pullback f g, pullback.snd f g ⁻¹ᵁ V)) := ⋯
hψ : Function.Bijective ⇑ψ
φ : pushout (app (fromNormalization f) U) (appLE g U V hVU) ⟶
  Γ(normalization (pullback.snd f g), normalizationPullback f g ⁻¹ᵁ W) := ⋯
⊢ pushout.inr (app (fromNormalization f) U) (appLE g U V hVU) ≫
      (pushout.congrHom ⋯ ⋯).hom ≫ pushoutSection ⋯ hVU ⋯ ⋯ ≫ app (normalizationPullback f g) W =
    pushout.inr (app (fromNormalization f) U) (appLE g U V hVU) ≫ φ

Tactic:
· dsimp [φ]
    simp only [Scheme.Hom.app_eq_appLE, colimit.ι_desc_assoc, span_left, PushoutCocone.mk_pt,
      PushoutCocone.mk_ι_app, Category.id_comp, Scheme.Hom.appLE_comp_appLE, eqToHom_op,
      Category.assoc, IsPushout.inl_isoPushout_inv_assoc]
    simp_rw [← Category.assoc, ← IsIso.comp_inv_eq]
    simp only [← Functor.map_inv, inv_eqToHom, Scheme.Hom.appLE_map, IsIso.Iso.inv_inv,
      Category.assoc]
    have : Mono (CommRingCat.ofHom (integralClosure Γ(Y, V)
        Γ(pullback f g, pullback.snd f g ⁻¹ᵁ V)).val.toRingHom) :=
      ConcreteCategory.mono_of_injective _ Subtype.val_injective
    rw [← cancel_mono (CommRingCat.ofHom (Subalgebra.val _).toRingHom)]
    simp only [Category.assoc, Scheme.Hom.normalizationObjIso_hom_val, Scheme.Hom.appLE_comp_appLE,
      Scheme.Hom.toNormalization_normalizationPullback_fst, ← CommRingCat.ofHom_comp]
    have H : pullback.snd f g ⁻¹ᵁ V ≤ pullback.fst f g ⁻¹ᵁ f ⁻¹ᵁ U := by
      rw [← Scheme.Hom.comp_preimage, pullback.condition, Scheme.Hom.comp_preimage]
      exact Scheme.Hom.preimage_mono _ hVU
    trans (f.normalizationObjIso hU).hom ≫ CommRingCat.ofHom
        (integralClosure Γ(S, U) Γ(X, f ⁻¹ᵁ U)).val.toRingHom ≫ (pullback.fst f g).appLE _ _ H
    · rw [reassoc_of% Scheme.Hom.normalizationObjIso_hom_val, Scheme.Hom.appLE_comp_appLE]
    · congr 1
      ext x
      change (pullback.fst f g).appLE _ _ H x = _
      trans (CommRingCat.ofHom Algebra.TensorProduct.includeRight.toRingHom ≫ e₀.hom) x
      · congr 2; simp [e₀, pushout.inl_desc_assoc]
      · simp [ψ, toIntegralClosure, e]; rfl

Goal state after:
case h₁
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
inst✝⁴ : QuasiCompact f✝
inst✝³ : QuasiSeparated f✝
X S Y : Scheme
f : X ⟶ S
g : Y ⟶ S
inst✝² : QuasiCompact f
inst✝¹ : QuasiSeparated f
inst✝ : Smooth g
x : ↥(pullback (fromNormalization f) g)
U : TopologicalSpace.Opens ↥S
hU : U ∈ S.affineOpens
hxU : (pullback.snd (fromNormalization f) g ≫ g) x ∈ ↑U
V : TopologicalSpace.Opens ↥Y
hV : V ∈ Y.affineOpens
hxV : (pullback.snd (fromNormalization f) g) x ∈ ↑V
hVU : V ≤ g ⁻¹ᵁ U
W : (pullback (fromNormalization f) g).Opens := ⋯
this✝¹ : IsIso (pushoutSection ⋯ hVU ⋯ ⋯)
this✝ : (CommRingCat.Hom.hom (appLE g U V hVU)).Smooth
algInst✝² : Algebra ↑Γ(S, U) ↑Γ(X, f ⁻¹ᵁ U) := ⋯
algInst✝¹ : Algebra ↑Γ(S, U) ↑Γ(Y, V) := ⋯
algInst✝ : Algebra ↑Γ(Y, V) ↑Γ(pullback f g, pullback.snd f g ⁻¹ᵁ V) := ⋯
algebraizeInst✝ : Algebra.Smooth ↑Γ(S, U) ↑Γ(Y, V)
this : IsIso (pushoutSection ⋯ hVU ⋯ ⋯)
e₀ : CommRingCat.of
    (↑(Y.presheaf.1 (Opposite.op V)) ⊗[↑(S.presheaf.1 (Opposite.op U))] ↑(X.presheaf.1 (Opposite.op (f ⁻¹ᵁ U)))) ≅
  Γ(pullback f g, pullback.snd f g ⁻¹ᵁ V) := ⋯
e : ↑Γ(Y, V) ⊗[↑Γ(S, U)] ↑Γ(X, f ⁻¹ᵁ U) ≃ₐ[↑Γ(Y, V)] ↑Γ(pullback f g, pullback.snd f g ⁻¹ᵁ V) := ⋯
ψ : ↑Γ(Y, V) ⊗[↑Γ(S, U)] ↥(integralClosure ↑Γ(S, U) ↑Γ(X, f ⁻¹ᵁ U)) →ₐ[↑Γ(Y, V)]
  ↥(integralClosure ↑Γ(Y, V) ↑Γ(pullback f g, pullback.snd f g ⁻¹ᵁ V)) := ⋯
hψ : Function.Bijective ⇑ψ
φ : pushout (app (fromNormalization f) U) (appLE g U V hVU) ⟶
  Γ(normalization (pullback.snd f g), normalizationPullback f g ⁻¹ᵁ W) := ⋯
⊢ pushout.inr (app (fromNormalization f) U) (appLE g U V hVU) ≫
      (pushout.congrHom ⋯ ⋯).hom ≫ pushoutSection ⋯ hVU ⋯ ⋯ ≫ app (normalizationPullback f g) W =
    pushout.inr (app (fromNormalization f) U) (appLE g U V hVU) ≫ φ