-- Module: Mathlib.RingTheory.Smooth.NoetherianDescent
Goal:
R : Type u_1
inst : CommRing R
A : Type u
B : Type u_2
inst_1 : CommRing A
inst_2 : Algebra R A
inst_3 : CommRing B
inst_4 : Algebra A B
inst_5 : Smooth A B
P : Presentation A B (Fin (Presentation.ofFinitePresentationVars A B))
  (Fin (Presentation.ofFinitePresentationRels A B)) :=
  Presentation.ofFinitePresentation A B
f : P.Ring →ₐ[A] B := IsScalarTower.toAlgHom A P.Ring B
hkerf : Eq (α := Ideal P.Ring) (RingHom.ker f) (Ideal.span (Set.range P.relation))
σ :
  B →ₐ[A]
    MvPolynomial (Fin (Presentation.ofFinitePresentationVars A B)) A ⧸
      HPow.hPow (α := Ideal P.Ring) (β := ℕ) (RingHom.ker f) 2
hsig_1 : f.kerSquareLift.comp σ = AlgHom.id A B
h :
  (i : Fin (Presentation.ofFinitePresentationVars A B)) →
    MvPolynomial (Fin (Presentation.ofFinitePresentationVars A B)) A
hh :
  ∀ (i : Fin (Presentation.ofFinitePresentationVars A B)),
    Eq (α :=
      MvPolynomial (Fin (Presentation.ofFinitePresentationVars A B)) A ⧸
        HPow.hPow (α := Ideal P.Ring) (β := ℕ) (RingHom.ker f) 2)
      ((Ideal.Quotient.mk (HPow.hPow (α := Ideal P.Ring) (β := ℕ) (RingHom.ker f) 2) :
            (a : MvPolynomial (Fin (Presentation.ofFinitePresentationVars A B)) A) →
              MvPolynomial (Fin (Presentation.ofFinitePresentationVars A B)) A ⧸ RingHom.ker f ^ 2)
          (h i) :
        MvPolynomial (Fin (Presentation.ofFinitePresentationVars A B)) A ⧸
          HPow.hPow (α := Ideal P.Ring) (β := ℕ) (RingHom.ker f) 2)
      ((σ : (a : B) → MvPolynomial (Fin (Presentation.ofFinitePresentationVars A B)) A ⧸ RingHom.ker f ^ 2) (P.val i) :
        MvPolynomial (Fin (Presentation.ofFinitePresentationVars A B)) A ⧸
          HPow.hPow (α := Ideal P.Ring) (β := ℕ) (RingHom.ker f) 2)
hdiag :
  AlgHom.comp (Ideal.Quotient.mkₐ A (HPow.hPow (α := Ideal P.Ring) (β := ℕ) (RingHom.ker f) 2)) (aeval h) =
    σ.comp (aeval P.val)
p :
  (j : Fin (Presentation.ofFinitePresentationRels A B)) →
    MvPolynomial (Fin (Presentation.ofFinitePresentationRels A B))
      (MvPolynomial (Fin (Presentation.ofFinitePresentationVars A B)) A)
homog : ∀ (j : Fin (Presentation.ofFinitePresentationRels A B)), (p j).IsHomogeneous 2
hp :
  ∀ (j : Fin (Presentation.ofFinitePresentationRels A B)),
    Eq (α := MvPolynomial (Fin (Presentation.ofFinitePresentationVars A B)) A)
      ((eval P.relation :
            (a :
                MvPolynomial (Fin (Presentation.ofFinitePresentationRels A B))
                  (MvPolynomial (Fin (Presentation.ofFinitePresentationVars A B)) A)) →
              MvPolynomial (Fin (Presentation.ofFinitePresentationVars A B)) A)
          (p j) :
        MvPolynomial (Fin (Presentation.ofFinitePresentationVars A B)) A)
      ((aeval h :
            (a : MvPolynomial (Fin (Presentation.ofFinitePresentationVars A B)) A) →
              MvPolynomial (Fin (Presentation.ofFinitePresentationVars A B)) A)
          (P.relation j) :
        MvPolynomial (Fin (Presentation.ofFinitePresentationVars A B)) A)
hsig : ∀ (i : Fin (Presentation.ofFinitePresentationVars A B)), (f : (a : P.Ring) → B) (h i) = P.val i
this :
  ∀ (i : Fin (Presentation.ofFinitePresentationVars A B)),
    ∃ (p : MvPolynomial (Fin (Presentation.ofFinitePresentationRels A B)) P.Ring),
      p.IsHomogeneous 1 ∧
        (eval P.relation : (a : MvPolynomial (Fin (Presentation.ofFinitePresentationRels A B)) P.Ring) → P.Ring) p =
          h i - X i
choose :
  ∀
    (q :
      (i : Fin (Presentation.ofFinitePresentationVars A B)) →
        MvPolynomial (Fin (Presentation.ofFinitePresentationRels A B)) P.Ring)
    (hqhom : ∀ (i : Fin (Presentation.ofFinitePresentationVars A B)), (q i).IsHomogeneous 1)
    (hq :
      ∀ (i : Fin (Presentation.ofFinitePresentationVars A B)),
        (eval P.relation : (a : MvPolynomial (Fin (Presentation.ofFinitePresentationRels A B)) P.Ring) → P.Ring) (q i) =
          h i - X i),
    ∃ (A₀ : Subalgebra R A) (B₀ : Type u) (x : CommRing B₀) (x_1 : Algebra (↥A₀) B₀),
      A₀.FG ∧ Smooth (↥A₀) B₀ ∧ Nonempty (B ≃ₐ[A] A ⊗[↥A₀] B₀)
⊢ ∃ (A₀ : Subalgebra R A) (B₀ : Type u) (x : CommRing B₀) (x_1 : Algebra (↥A₀) B₀),
    A₀.FG ∧ Smooth (↥A₀) B₀ ∧ Nonempty (B ≃ₐ[A] A ⊗[↥A₀] B₀)

Tactic:
choose q hqhom hq using this