import Mathlib
import OAI.AlgebraicGeometry.Seshadri.Projective.SectionMaps

namespace OAI

section
noncomputable section
noncomputable section
open CategoryTheory
open CategoryTheory.Category CategoryTheory.Functor
universe v u v₁ v₂ u₁ u₂
namespace MaximalSeshadri.Projective
noncomputable section
open AlgebraicGeometry CategoryTheory CategoryTheory.Limits TopologicalSpace
open scoped AlgebraicGeometry
open MvPolynomial HomogeneousLocalization
variable {K σ R : Type u} [CommRing K] [CommRing R]
attribute [local instance] MvPolynomial.gradedAlgebra

lemma polynomial_irrelevant_le :
    (HomogeneousIdeal.irrelevant (homogeneousSubmodule σ K)).toIdeal ≤
      Ideal.span (Set.range (MvPolynomial.X (R := K) : σ → MvPolynomial σ K)) := by
  rw [HomogeneousIdeal.toIdeal_irrelevant_le]
  intro n hn p hp
  change p ∈ idealOfVars σ K
  rw [← pow_one (idealOfVars σ K), mem_pow_idealOfVars_iff]
  intro d hd
  change 1 ≤ d.sum fun _ v => v
  rw [Finsupp.sum, ← hp.degree_eq_sum_deg_support hd]
  exact hn

lemma projective_coordinate_cover :
    (⨆ i : σ, Proj.basicOpen (homogeneousSubmodule σ K) (X i)) = ⊤ :=
  Proj.iSup_basicOpen_eq_top _ _ polynomial_irrelevant_le

def projectiveCoordinateCover : (Proj (homogeneousSubmodule σ K)).AffineOpenCover :=
  Proj.affineOpenCoverOfIrrelevantLESpan _ (X (R := K))
    (m := fun _ => 1) (fun i => isHomogeneous_X K i) (fun _ => by decide)
    polynomial_irrelevant_le

lemma normalized_evalAway_surjective (k : K →+* R) (a : σ → R)
    (i : σ) (hi : a i = 1) (h : Function.Surjective (eval₂Hom k a)) :
    Function.Surjective (evalAway (𝒜 := homogeneousSubmodule σ K) (eval₂Hom k a)
      (X i) (by simpa only [eval₂Hom_X', hi] using (isUnit_one : IsUnit (1 : R)))) := by
  let F := evalAway (𝒜 := homogeneousSubmodule σ K) (eval₂Hom k a)
    (X i) (by simpa only [eval₂Hom_X', hi] using (isUnit_one : IsUnit (1 : R)))
  have hC (r : K) : C r ∈ homogeneousSubmodule σ K (0 • 1) := by simp
  have hX (j : σ) : X j ∈ homogeneousSubmodule σ K (1 • 1) := by
    simpa using isHomogeneous_X K j
  let c : K →+* Away (homogeneousSubmodule σ K) (X i) :=
    (HomogeneousLocalization.fromZeroRingHom _ _).comp
      { toFun := fun r => ⟨C r, isHomogeneous_C _ _⟩
        map_one' := Subtype.ext C_1
        map_mul' := fun _ _ => Subtype.ext C_mul
        map_zero' := Subtype.ext C_0
        map_add' := fun _ _ => Subtype.ext C_add }
  let b (j : σ) := Away.mk _ (isHomogeneous_X K i) 1 (X j) (hX j)
  have hc (r : K) : F (c r) = k r := by
    change F (Away.mk _ (isHomogeneous_X K i) 0 (C r) (hC r)) = k r
    have H := evalAway_mk_clear (eval₂Hom k a) (isHomogeneous_X K i)
      (show IsUnit ((eval₂Hom k a) (X i)) by simpa only [eval₂Hom_X', hi] using
        (isUnit_one : IsUnit (1 : R))) 0 (C r) (hC r)
    simpa only [F, eval₂Hom_C, eval₂Hom_X', pow_zero, mul_one] using H
  have hb (j : σ) : F (b j) = a j := by
    have H := evalAway_mk_clear (eval₂Hom k a) (isHomogeneous_X K i)
      (show IsUnit ((eval₂Hom k a) (X i)) by simpa only [eval₂Hom_X', hi] using
        (isUnit_one : IsUnit (1 : R))) 1 (X j) (hX j)
    simpa only [F, b, eval₂Hom_X', hi, one_pow, mul_one] using H
  have H : F.comp (eval₂Hom c b) = eval₂Hom k a := by
    apply MvPolynomial.ringHom_ext
    · intro r
      simpa only [RingHom.comp_apply, eval₂Hom_C] using hc r
    · intro j
      simpa only [RingHom.comp_apply, eval₂Hom_X'] using hb j
  intro r
  obtain ⟨p, hp⟩ := h r
  exact ⟨eval₂Hom c b p, (RingHom.congr_fun H p).trans hp⟩

lemma normalized_coordinates_closed {X : Scheme.{u}} [IsAffine X]
    (k : K →+* Γ(X, ⊤)) (a : σ → Γ(X, ⊤)) (i : σ) (hi : a i = 1)
    (h : Function.Surjective (eval₂Hom k a)) :
    IsClosedImmersion (X.toSpecΓ ≫ Spec.map (CommRingCat.ofHom
      (evalAway (𝒜 := homogeneousSubmodule σ K) (eval₂Hom k a) (MvPolynomial.X i)
        (by simpa only [eval₂Hom_X', hi] using (isUnit_one : IsUnit (1 : Γ(X, ⊤))))))) := by
  have := IsClosedImmersion.spec_of_surjective
    (CommRingCat.ofHom (evalAway (𝒜 := homogeneousSubmodule σ K) (eval₂Hom k a)
      (MvPolynomial.X i) (by simpa only [eval₂Hom_X', hi] using
        (isUnit_one : IsUnit (1 : Γ(X, ⊤))))))
    (normalized_evalAway_surjective k a i hi h)
  infer_instance

end
end MaximalSeshadri.Projective
namespace MaximalSeshadri.Projective
noncomputable section
open AlgebraicGeometry CategoryTheory CategoryTheory.Limits TopologicalSpace
open scoped AlgebraicGeometry
open MaximalSeshadri.Frames MvPolynomial
variable {K σ : Type u} [CommRing K] {X : Scheme.{u}}
attribute [local instance] MvPolynomial.gradedAlgebra

theorem sectionsMorphism_isClosedImmersion {M : X.Modules} (k : K →+* Γ(X, ⊤))
    (s : σ → (O X ⟶ M)) (hcover : (⨆ i, SectionOpens.isoOpen (s i)) = ⊤)
    (haff : ∀ i, IsAffine (SectionOpens.isoOpen (s i)).toScheme)
    (hgen : ∀ i, Function.Surjective
      (eval₂Hom (((SectionOpens.isoOpen (s i)).ι.appTop.hom).comp k)
        (fun j => coefficient (sectionFrame (s i))
          (restrictSection (SectionOpens.isoOpen (s i)).ι (s j))))) :
    IsClosedImmersion (sectionsMorphism k s hcover) := by
  let V := projectiveCoordinateCover (K := K) (σ := σ)
  apply IsZariskiLocalAtTarget.of_openCover (P := @IsClosedImmersion) V.openCover
  intro i
  change σ at i
  let U := SectionOpens.isoOpen (s i)
  let : IsAffine U.toScheme := haff i
  let k' := U.ι.appTop.hom.comp k
  let a : σ → Γ(U.toScheme, ⊤) := fun j => coefficient (sectionFrame (s i)) (restrictSection U.ι (s j))
  have hi : a i = 1 := sectionFrame_normalized (s i)
  let g := U.toScheme.toSpecΓ ≫ Spec.map (CommRingCat.ofHom
    (evalAway (𝒜 := homogeneousSubmodule σ K) (eval₂Hom k' a) (MvPolynomial.X i)
      (by simpa only [eval₂Hom_X', hi] using (isUnit_one : IsUnit (1 : Γ(U.toScheme, ⊤))))))
  have hg : IsClosedImmersion g := normalized_coordinates_closed k' a i hi (hgen i)
  let chart := Proj.awayι (homogeneousSubmodule σ K) (MvPolynomial.X i)
    (isHomogeneous_X K i) (show 0 < 1 by decide)
  have : IsOpenImmersion chart := V.map_prop i
  have H : IsPullback g U.ι chart (sectionsMorphism k s hcover) := by
    apply IsOpenImmersion.isPullback
    · exact (sectionsMorphism_local k s hcover i).trans (Category.assoc _ _ _).symm
    · have hrange : chart.opensRange =
          Proj.basicOpen (homogeneousSubmodule σ K) (MvPolynomial.X i) :=
        Proj.opensRange_awayι _ _ (isHomogeneous_X K i) (show 0 < 1 by decide)
      rw [hrange, sectionsMorphism_preimage, Scheme.Opens.opensRange_ι]
  change IsClosedImmersion (pullback.snd (sectionsMorphism k s hcover) chart)
  rw [← H.flip.isoPullback_inv_snd]
  infer_instance

end
end MaximalSeshadri.Projective

namespace MaximalSeshadri.Geometry
noncomputable section
open CategoryTheory AlgebraicGeometry TopologicalSpace
open scoped AlgebraicGeometry
variable {X Y : Scheme.{u}} {ι : Type u}

theorem closedImmersion_of_source_charts (f : X ⟶ Y) [UniversallyClosed f]
    (U : ι → X.Opens) (V : ι → Y.Opens)
    (hU : (⨆ i, U i) = ⊤) (g : ∀ i, (U i).toScheme ⟶ (V i).toScheme)
    (hg : ∀ i, IsClosedImmersion (g i))
    (hcomm : ∀ i, (U i).ι ≫ f = g i ≫ (V i).ι)
    (hpre : ∀ i, f ⁻¹ᵁ V i = U i) : IsClosedImmersion f := by
  have hinj : Function.Injective f := by
    intro x y hxy
    have hx : x ∈ ⨆ i, U i := by rw [hU]; trivial
    obtain ⟨i, hi⟩ := Opens.mem_iSup.mp hx
    have hy : y ∈ U i := by
      rw [← hpre i]
      change f y ∈ V i
      rw [← hxy]
      change x ∈ f ⁻¹ᵁ V i
      rw [hpre]
      exact hi
    have he (z : (U i).toScheme) : (g i z).val = f z.val :=
      (congrArg (fun a : (U i).toScheme ⟶ Y => a z) (hcomm i)).symm
    have hz : g i ⟨x, hi⟩ = g i ⟨y, hy⟩ := by
      apply Subtype.ext
      exact (he ⟨x, hi⟩).trans (hxy.trans (he ⟨y, hy⟩).symm)
    have := (g i).isClosedEmbedding.injective hz
    exact congrArg Subtype.val this
  have hstalk : SurjectiveOnStalks f := by
    apply IsZariskiLocalAtSource.of_openCover (P := @SurjectiveOnStalks)
      (X.openCoverOfIsOpenCover U hU)
    intro i
    change ι at i
    change SurjectiveOnStalks ((U i).ι ≫ f)
    rw [hcomm]
    have := hg i
    infer_instance
  exact ⟨Topology.IsClosedEmbedding.of_continuous_injective_isClosedMap f.continuous hinj f.isClosedMap⟩

end
end MaximalSeshadri.Geometry

namespace MaximalSeshadri.Projective
noncomputable section
open AlgebraicGeometry CategoryTheory HomogeneousLocalization MvPolynomial
open scoped AlgebraicGeometry
variable {K σ : Type u} [CommRing K]
attribute [local instance] MvPolynomial.gradedAlgebra

def projectiveBase : Proj (homogeneousSubmodule σ K) ⟶ Spec (CommRingCat.of K) :=
  Proj.toSpecZero _ ≫ Spec.map (CommRingCat.ofHom (projectiveConstants (σ := σ)))

lemma evalAway_constants {R : Type u} [CommRing R] (k : K →+* R) (a : σ → R)
    (i : σ) (hi : a i = 1) :
    (evalAway (𝒜 := homogeneousSubmodule σ K) (eval₂Hom k a) (X i)
      (by simpa only [eval₂Hom_X', hi] using (isUnit_one : IsUnit (1 : R)))).comp
      ((fromZeroRingHom (homogeneousSubmodule σ K) (.powers (X i))).comp
        projectiveConstants) = k := by
  apply RingHom.ext
  intro c
  change IsLocalization.Away.lift (X i)
    (by simpa only [eval₂Hom_X', hi] using (isUnit_one : IsUnit (1 : R)))
    (HomogeneousLocalization.fromZeroRingHom (homogeneousSubmodule σ K)
      (.powers (X i)) (projectiveConstants c)).val = k c
  change IsLocalization.Away.lift (X i)
    (by simpa only [eval₂Hom_X', hi] using (isUnit_one : IsUnit (1 : R)))
    (algebraMap (MvPolynomial σ K) (Localization.Away (X i)) (C c)) = k c
  rw [IsLocalization.Away.lift_eq, eval₂Hom_C]

@[reassoc]
theorem coordinatesMap_over {X : Scheme.{u}} (k : K →+* Γ(X, ⊤))
    (a : σ → Γ(X, ⊤)) (i : σ) (hi : a i = 1) :
    coordinatesMap X k a i hi ≫ projectiveBase =
      X.toSpecΓ ≫ Spec.map (CommRingCat.ofHom k) := by
  simp only [coordinatesMap, fromUnitCoordinate, projectiveBase, Category.assoc,
    Proj.awayι_toSpecZero_assoc, ← Spec.map_comp, ← CommRingCat.ofHom_comp]
  congr 2
  exact congrArg CommRingCat.ofHom (evalAway_constants k a i hi)

@[reassoc]
theorem sectionsMorphism_over {X : Scheme.{u}} {M : X.Modules}
    (k : K →+* Γ(X, ⊤)) (s : σ → (MaximalSeshadri.Frames.O X ⟶ M))
    (hcover : (⨆ i, SectionOpens.isoOpen (s i)) = ⊤) :
    sectionsMorphism k s hcover ≫ projectiveBase =
      X.toSpecΓ ≫ Spec.map (CommRingCat.ofHom k) := by
  apply (X.openCoverOfIsOpenCover (fun i => SectionOpens.isoOpen (s i)) hcover).hom_ext
  intro i
  change σ at i
  change (SectionOpens.isoOpen (s i)).ι ≫ _ = _
  rw [← Category.assoc, sectionsMorphism_local, coordinatesMap_over]
  change _ = (SectionOpens.isoOpen (s i)).ι ≫ X.toSpecΓ ≫ _
  rw [← Category.assoc, Scheme.toSpecΓ_naturality, Category.assoc, ← Spec.map_comp]
  rfl

end
end MaximalSeshadri.Projective

namespace MaximalSeshadri.Projective
noncomputable section
open AlgebraicGeometry CategoryTheory TopologicalSpace MvPolynomial
open scoped AlgebraicGeometry
open MaximalSeshadri.Frames
variable {K σ ι : Type u} [CommRing K] {X : Scheme.{u}}
attribute [local instance] MvPolynomial.gradedAlgebra

theorem sectionsMorphism_closed_on_cover {M : X.Modules} (k : K →+* Γ(X, ⊤))
    (s : σ → (O X ⟶ M)) (hcover : (⨆ i, SectionOpens.isoOpen (s i)) = ⊤)
    [UniversallyClosed (sectionsMorphism k s hcover)]
    (a : ι → σ) (hsub : (⨆ i, SectionOpens.isoOpen (s (a i))) = ⊤)
    (haff : ∀ i, IsAffine (SectionOpens.isoOpen (s (a i))).toScheme)
    (hgen : ∀ i, Function.Surjective
      (eval₂Hom (((SectionOpens.isoOpen (s (a i))).ι.appTop.hom).comp k)
        (fun j => coefficient (sectionFrame (s (a i)))
          (restrictSection (SectionOpens.isoOpen (s (a i))).ι (s j))))) :
    IsClosedImmersion (sectionsMorphism k s hcover) := by
  let U (i : ι) := SectionOpens.isoOpen (s (a i))
  let V (i : ι) := Proj.basicOpen (homogeneousSubmodule σ K) (MvPolynomial.X (a i))
  let k' (i : ι) := (U i).ι.appTop.hom.comp k
  let b (i : ι) (j : σ) := coefficient (sectionFrame (s (a i)))
    (restrictSection (U i).ι (s j))
  have hb (i : ι) : b i (a i) = 1 := sectionFrame_normalized (s (a i))
  let g (i : ι) := (U i).toScheme.toSpecΓ ≫ Spec.map (CommRingCat.ofHom
    (evalAway (𝒜 := homogeneousSubmodule σ K) (eval₂Hom (k' i) (b i)) (MvPolynomial.X (a i))
      (by simpa only [eval₂Hom_X', hb] using (isUnit_one : IsUnit (1 : Γ((U i).toScheme, ⊤))))))
  let e (i : ι) := Proj.basicOpenIsoSpec (homogeneousSubmodule σ K) (MvPolynomial.X (a i))
    (isHomogeneous_X K (a i)) (by decide : 0 < (1 : ℕ))
  apply MaximalSeshadri.Geometry.closedImmersion_of_source_charts
    (sectionsMorphism k s hcover) U V hsub (fun i => g i ≫ (e i).inv)
  · intro i
    let : IsAffine (U i).toScheme := haff i
    have : IsClosedImmersion (g i) := normalized_coordinates_closed (k' i) (b i) (a i) (hb i) (hgen i)
    infer_instance
  · intro i
    rw [sectionsMorphism_local]
    rfl
  · intro i
    exact sectionsMorphism_preimage k s hcover (a i)

end
end MaximalSeshadri.Projective

namespace MaximalSeshadri.Projective
noncomputable section
open AlgebraicGeometry CategoryTheory TopologicalSpace MvPolynomial
open scoped AlgebraicGeometry
open MaximalSeshadri.Frames
variable {K σ : Type u} [CommRing K] {X : Scheme.{u}}
attribute [local instance] MvPolynomial.gradedAlgebra

instance projectiveBase_separated : IsSeparated (projectiveBase (K := K) (σ := σ)) := by
  unfold projectiveBase
  infer_instance

instance sectionsMorphism_proper {M : X.Modules} (k : K →+* Γ(X, ⊤))
    (s : σ → (O X ⟶ M)) (hcover : (⨆ i, SectionOpens.isoOpen (s i)) = ⊤)
    [IsProper (X.toSpecΓ ≫ Spec.map (CommRingCat.ofHom k))] :
    IsProper (sectionsMorphism k s hcover) := by
  have : IsProper (sectionsMorphism k s hcover ≫ projectiveBase) := by
    rw [sectionsMorphism_over]
    infer_instance
  exact IsProper.of_comp _ projectiveBase

lemma toSpec_scalarMap (p : X ⟶ Spec (CommRingCat.of K)) :
    X.toSpecΓ ≫ Spec.map (CommRingCat.ofHom
      (p.appTop.hom.comp (Scheme.ΓSpecIso (CommRingCat.of K)).inv.hom)) = p := by
  change X.toSpecΓ ≫ Spec.map ((Scheme.ΓSpecIso (CommRingCat.of K)).inv ≫ p.appTop) = p
  rw [Spec.map_comp, ← Category.assoc, ← Scheme.toSpecΓ_naturality]
  simp

end
end MaximalSeshadri.Projective

namespace MaximalSeshadri.Geometry
noncomputable section
open AlgebraicGeometry CategoryTheory MvPolynomial
open scoped AlgebraicGeometry

lemma finiteType_coordinates {K R : Type u} [CommRing K] [CommRing R]
    (k : K →+* R) (hk : k.FiniteType) :
    ∃ n : ℕ, ∃ a : Fin n → R, Function.Surjective (eval₂Hom k a) := by
  let := k.toAlgebra
  have : Algebra.FiniteType K R := hk
  obtain ⟨n, f, hf⟩ := Algebra.FiniteType.iff_quotient_mvPolynomial''.mp
    (inferInstance : Algebra.FiniteType K R)
  refine ⟨n, fun i => f (X i), ?_⟩
  have heq : eval₂Hom k (fun i => f (X i)) = f.toRingHom := by
    apply MvPolynomial.ringHom_ext
    · intro c
      exact (eval₂Hom_C _ _ _).trans (f.commutes c).symm
    · intro i
      exact eval₂Hom_X' _ _ _
  rw [heq]
  exact hf

theorem affine_coordinate_generators {K : Type u} [CommRing K] {X : Scheme.{u}}
    [IsAffine X] (p : X ⟶ Spec (CommRingCat.of K)) [LocallyOfFiniteType p] :
    ∃ n : ℕ, ∃ a : Fin n → Γ(X, ⊤), Function.Surjective
      (eval₂Hom (p.appTop.hom.comp (Scheme.ΓSpecIso (CommRingCat.of K)).inv.hom) a) := by
  have hp : p.appTop.hom.FiniteType :=
    HasRingHomProperty.appTop (P := @LocallyOfFiniteType) _ inferInstance
  have he : (Scheme.ΓSpecIso (CommRingCat.of K)).inv.hom.FiniteType :=
    RingHom.FiniteType.of_surjective _ (ConcreteCategory.bijective_of_isIso
      (Scheme.ΓSpecIso (CommRingCat.of K)).inv).surjective
  exact finiteType_coordinates _ (hp.comp he)

end
end MaximalSeshadri.Geometry


end
end
end

end OAI