-- Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Scheme.lean
/--
If `x` is a point in the basic open set `D(f)` where `f` is a homogeneous element of positive
degree, then the homogeneously localized ring `A⁰ₓ` has the universal property of the localization
of `A⁰_f` at `φ(x)` where `φ : Proj|D(f) ⟶ Spec A⁰_f` is the morphism of locally ringed space
constructed as above.
-/
lemma isLocalization_atPrime (f) (x : pbo f) {m} (f_deg : f ∈ 𝒜 m) (hm : 0 < m) :
    @IsLocalization (Away 𝒜 f) _ ((toSpec 𝒜 f).base x).asIdeal.primeCompl
      (AtPrime 𝒜 x.1.asHomogeneousIdeal.toIdeal) _
      (mapId 𝒜 (Submonoid.powers_le.mpr x.2)).toAlgebra

Proof state:
case intro.mk.intro
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
x : ↥(pbo f)
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
this : Algebra (A⁰_ f) (AtPrime 𝒜 (↑x).asHomogeneousIdeal.toIdeal) := (mapId 𝒜 ⋯).toAlgebra
y z : NumDenSameDeg 𝒜 (Submonoid.powers f)
e :
  (algebraMap A (Localization (↑x).asHomogeneousIdeal.toIdeal.primeCompl)) (↑z.den * ↑y.num) =
    (algebraMap A (Localization (↑x).asHomogeneousIdeal.toIdeal.primeCompl)) (↑y.den * ↑z.num)
c : A
hcx : c ∈ (↑x).asHomogeneousIdeal.toIdeal.primeCompl
i : ℕ
hi : ↑(((decompose 𝒜) c) i) ∉ (↑x).asHomogeneousIdeal.toIdeal
hc :
  ↑(((decompose 𝒜) (↑⟨c, hcx⟩ * (↑z.den * ↑y.num))) (i + z.deg + y.deg)) =
    ↑(((decompose 𝒜) (↑⟨c, hcx⟩ * (↑y.den * ↑z.num))) (i + z.deg + y.deg))
⊢ ↑(((decompose 𝒜) c) i) * (↑z.den * ↑y.num) = ↑(((decompose 𝒜) c) i) * (↑y.den * ↑z.num)

Tactic:
conv_rhs at hc => rw [add_right_comm]

Result:
case intro.mk.intro
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
x : ↥(pbo f)
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
this : Algebra (A⁰_ f) (AtPrime 𝒜 (↑x).asHomogeneousIdeal.toIdeal) := (mapId 𝒜 ⋯).toAlgebra
y z : NumDenSameDeg 𝒜 (Submonoid.powers f)
e :
  (algebraMap A (Localization (↑x).asHomogeneousIdeal.toIdeal.primeCompl)) (↑z.den * ↑y.num) =
    (algebraMap A (Localization (↑x).asHomogeneousIdeal.toIdeal.primeCompl)) (↑y.den * ↑z.num)
c : A
hcx : c ∈ (↑x).asHomogeneousIdeal.toIdeal.primeCompl
i : ℕ
hi : ↑(((decompose 𝒜) c) i) ∉ (↑x).asHomogeneousIdeal.toIdeal
hc :
  ↑(((decompose 𝒜) (↑⟨c, hcx⟩ * (↑z.den * ↑y.num))) (i + z.deg + y.deg)) =
    ↑(((decompose 𝒜) (↑⟨c, hcx⟩ * (↑y.den * ↑z.num))) (i + y.deg + z.deg))
⊢ ↑(((decompose 𝒜) c) i) * (↑z.den * ↑y.num) = ↑(((decompose 𝒜) c) i) * (↑y.den * ↑z.num)

Proof state:
case surj'.intro.mk.mk.refine_2
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
x : ↥(pbo f)
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
this : Algebra (A⁰_ f) (AtPrime 𝒜 (↑x).asHomogeneousIdeal.toIdeal) := (mapId 𝒜 ⋯).toAlgebra
i : ℕ
a : ↥(𝒜 i)
b : A
hb : b ∈ 𝒜 i
hb' : b ∉ (↑x).asHomogeneousIdeal
⊢ HomogeneousLocalization.val
      (HomogeneousLocalization.mk { deg := i, num := a, den := ⟨b, hb⟩, den_mem := hb' } *
        (algebraMap (A⁰_ f) (AtPrime 𝒜 (↑x).asHomogeneousIdeal.toIdeal))
          ↑(HomogeneousLocalization.mk { deg := i * m, num := ⟨↑a * b ^ (m - 1), ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ },
                ⟨HomogeneousLocalization.mk { deg := i * m, num := ⟨b ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ },
                  ⋯⟩).2) =
    HomogeneousLocalization.val
      ((algebraMap (A⁰_ f) (AtPrime 𝒜 (↑x).asHomogeneousIdeal.toIdeal))
        (HomogeneousLocalization.mk { deg := i * m, num := ⟨↑a * b ^ (m - 1), ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ },
            ⟨HomogeneousLocalization.mk { deg := i * m, num := ⟨b ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ }, ⋯⟩).1)

Tactic:
simp only [RingHom.algebraMap_toAlgebra, map_mk, RingHom.id_apply, val_mul, val_mk,
  mk_eq_mk', ← IsLocalization.mk'_mul, Submonoid.mk_mul_mk, IsLocalization.mk'_eq_iff_eq]

Result:
case surj'.intro.mk.mk.refine_2
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
x : ↥(pbo f)
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
this : Algebra (A⁰_ f) (AtPrime 𝒜 (↑x).asHomogeneousIdeal.toIdeal) := (mapId 𝒜 ⋯).toAlgebra
i : ℕ
a : ↥(𝒜 i)
b : A
hb : b ∈ 𝒜 i
hb' : b ∉ (↑x).asHomogeneousIdeal
⊢ (algebraMap A (Localization (↑x).asHomogeneousIdeal.toIdeal.primeCompl)) (f ^ i * (↑a * b ^ m)) =
    (algebraMap A (Localization (↑x).asHomogeneousIdeal.toIdeal.primeCompl)) (b * f ^ i * (↑a * b ^ (m - 1)))

Proof state:
case map_units'.mk
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
x : ↥(pbo f)
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
this : Algebra (A⁰_ f) (AtPrime 𝒜 (↑x).asHomogeneousIdeal.toIdeal) := (mapId 𝒜 ⋯).toAlgebra
y : A⁰_ f
hy : y ∈ ((ConcreteCategory.hom (toSpec 𝒜 f).base) x).asIdeal.primeCompl
⊢ IsUnit ((algebraMap (A⁰_ f) (AtPrime 𝒜 (↑x).asHomogeneousIdeal.toIdeal)) ↑⟨y, hy⟩)

Tactic:
obtain ⟨y, rfl⟩ := HomogeneousLocalization.mk_surjective y

Result:
case map_units'.mk.intro
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
x : ↥(pbo f)
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
this : Algebra (A⁰_ f) (AtPrime 𝒜 (↑x).asHomogeneousIdeal.toIdeal) := (mapId 𝒜 ⋯).toAlgebra
y : NumDenSameDeg 𝒜 (Submonoid.powers f)
hy : HomogeneousLocalization.mk y ∈ ((ConcreteCategory.hom (toSpec 𝒜 f).base) x).asIdeal.primeCompl
⊢ IsUnit ((algebraMap (A⁰_ f) (AtPrime 𝒜 (↑x).asHomogeneousIdeal.toIdeal)) ↑⟨HomogeneousLocalization.mk y, hy⟩)

Proof state:
case surj'.intro.mk.mk.refine_2
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
x : ↥(pbo f)
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
this : Algebra (A⁰_ f) (AtPrime 𝒜 (↑x).asHomogeneousIdeal.toIdeal) := (mapId 𝒜 ⋯).toAlgebra
i : ℕ
a : ↥(𝒜 i)
b : A
hb : b ∈ 𝒜 i
hb' : b ∉ (↑x).asHomogeneousIdeal
⊢ (algebraMap A (Localization (↑x).asHomogeneousIdeal.toIdeal.primeCompl)) (f ^ i * (↑a * b ^ m)) =
    (algebraMap A (Localization (↑x).asHomogeneousIdeal.toIdeal.primeCompl)) (b * f ^ i * (↑a * b ^ (m - 1)))

Tactic:
rw [mul_comm b, mul_mul_mul_comm, ← pow_succ', mul_assoc, tsub_add_cancel_of_le (by omega)]

Result:
no goals

Proof state:
case exists_of_eq.intro
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
x : ↥(pbo f)
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
this : Algebra (A⁰_ f) (AtPrime 𝒜 (↑x).asHomogeneousIdeal.toIdeal) := (mapId 𝒜 ⋯).toAlgebra
z : A⁰_ f
y : NumDenSameDeg 𝒜 (Submonoid.powers f)
e :
  (algebraMap (A⁰_ f) (AtPrime 𝒜 (↑x).asHomogeneousIdeal.toIdeal)) (HomogeneousLocalization.mk y) =
    (algebraMap (A⁰_ f) (AtPrime 𝒜 (↑x).asHomogeneousIdeal.toIdeal)) z
⊢ ∃ c, ↑c * HomogeneousLocalization.mk y = ↑c * z

Tactic:
obtain ⟨z, rfl⟩ := HomogeneousLocalization.mk_surjective z

Result:
case exists_of_eq.intro.intro
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
x : ↥(pbo f)
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
this : Algebra (A⁰_ f) (AtPrime 𝒜 (↑x).asHomogeneousIdeal.toIdeal) := (mapId 𝒜 ⋯).toAlgebra
y z : NumDenSameDeg 𝒜 (Submonoid.powers f)
e :
  (algebraMap (A⁰_ f) (AtPrime 𝒜 (↑x).asHomogeneousIdeal.toIdeal)) (HomogeneousLocalization.mk y) =
    (algebraMap (A⁰_ f) (AtPrime 𝒜 (↑x).asHomogeneousIdeal.toIdeal)) (HomogeneousLocalization.mk z)
⊢ ∃ c, ↑c * HomogeneousLocalization.mk y = ↑c * HomogeneousLocalization.mk z