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import OAI.AlgebraicGeometry.SectionFields.NormalDivisors
import OAI.AlgebraicGeometry.SectionFields.TensorAlgebra
namespace OAI
/-! Constants and function fields under geometric integrality and open charts. -/
noncomputable section
open AlgebraicGeometry CategoryTheory CategoryTheory.Limits TopologicalSpace Order Polynomial
open scoped TensorProduct WithZero
universe u
namespace RelativeDenominators
theorem global_constants_surjective_of_geometricallyIntegral
(K : Type u) [Field K] {X : Scheme.{u}}
(f : X ⟶ Spec (.of K)) [UniversallyClosed f] [GeometricallyIntegral f] :
Function.Surjective (((Scheme.ΓSpecIso (.of K)).inv ≫ f.appTop).hom) := by
let : IsIntegral X := GeometricallyIntegral.isIntegral_of_subsingleton f
let : CompactSpace X := (quasiCompact_iff_compactSpace f).mp inferInstance
let A := Γ(X, ⊤)
let : Field A := (isField_of_universallyClosed K f).toField
let F : CommRingCat.of K ⟶ A := (Scheme.ΓSpecIso (.of K)).inv ≫ f.appTop
let : Algebra K A := F.hom.toAlgebra
have hintegral : F.hom.IsIntegral := by
apply RingHom.isIntegral_respectsIso.2
(e := (Scheme.ΓSpecIso (.of K)).symm.commRingCatIsoToRingEquiv)
exact isIntegral_appTop_of_universallyClosed f
let : Algebra.IsIntegral K A := ⟨hintegral⟩
let g : Spec A ⟶ Spec (.of K) := Spec.map F
have hfactor : X.toSpecΓ ≫ g = f := by
dsimp [g, F]
rw [Spec.map_comp, ← Category.assoc, ← Scheme.toSpecΓ_naturality]
rw [Category.assoc, ← Scheme.isoSpec_Spec_inv]
simp
let P := pullback g g
let q : pullback X.toSpecΓ (pullback.fst g g) ⟶ P :=
pullback.snd X.toSpecΓ (pullback.fst g g)
let e : pullback X.toSpecΓ (pullback.fst g g) ≅ pullback f g :=
pullbackRightPullbackFstIso g g X.toSpecΓ ≪≫ pullback.congrHom hfactor rfl
let : IsIntegral (pullback X.toSpecΓ (pullback.fst g g)) :=
IsIntegral.of_isIso e.inv
let : IsSchemeTheoreticallyDominant X.toSpecΓ :=
IsSchemeTheoreticallyDominant.of_isDominant _
let : IsReduced P := IsSchemeTheoreticallyDominant.isReduced q
have hP : IsIrreducible (Set.univ : Set P) := by
have h := (IrreducibleSpace.isIrreducible_univ (pullback X.toSpecΓ (pullback.fst g g) : Scheme)).image
q q.continuous.continuousOn
simpa only [Set.image_univ, q.denseRange.closure_range] using h.closure
let : IrreducibleSpace P := (irreducibleSpace_def P).mpr hP
let : IsIntegral P := isIntegral_of_irreducibleSpace_of_isReduced P
let : IsIntegral (Spec (.of (A ⊗[K] A))) :=
IsIntegral.of_isIso (pullbackSpecIso K A A).hom
let : IsDomain (A ⊗[K] A) := (affine_isIntegral_iff (.of (A ⊗[K] A))).mp inferInstance
exact algebraMap_surjective_of_isAlgebraic_tensorSelf_domain K A
noncomputable def constantToFunctionField
(K : Type u) [Field K] {X : Scheme.{u}} [IsIntegral X]
(f : X ⟶ Spec (.of K)) : K →+* X.functionField :=
(X.germToFunctionField ⊤).hom.comp
(((Scheme.ΓSpecIso (.of K)).inv ≫ f.appTop).hom)
theorem rational_function_constant_of_divisor_zero
(K : Type u) [Field K] {X : Scheme.{u}} [IsIntegral X] [IsNoetherian X]
(f : X ⟶ Spec (.of K)) [UniversallyClosed f] [GeometricallyIntegral f]
(hNormal : ∀ x : X, IsIntegrallyClosed (X.presheaf.stalk x))
(u : X.functionFieldˣ) (hdiv : principalDivisor X u = 0) :
∃ c : Kˣ, constantToFunctionField K f (c : K) = (u : X.functionField) := by
obtain ⟨s, hs⟩ := rational_function_eq_global_unit_of_divisor_zero X hNormal u hdiv
obtain ⟨c, hc⟩ := global_constants_surjective_of_geometricallyIntegral K f (s : Γ(X, ⊤))
have hc0 : c ≠ 0 := by
intro h
apply Units.ne_zero s
rw [← hc, h, map_zero]
refine ⟨Units.mk0 c hc0, ?_⟩
change X.germToFunctionField ⊤
(((Scheme.ΓSpecIso (.of K)).inv ≫ f.appTop) c) = (u : X.functionField)
rw [hc]
exact hs
theorem functionFieldPullback_bijective_of_open
{X Y : Scheme} [IsIntegral X] [IsIntegral Y]
(j : Y ⟶ X) [IsOpenImmersion j] [IsDominant j] :
Function.Bijective (functionFieldPullback j) := by
let e := X.presheaf.stalkCongr
(Inseparable.of_eq (genericPoint_eq_of_dominant j))
have he : X.presheaf.stalkSpecializes
(specializes_of_eq (genericPoint_eq_of_dominant j)) = e.inv := rfl
have : IsIso (X.presheaf.stalkSpecializes
(specializes_of_eq (genericPoint_eq_of_dominant j))) := by rw [he]; infer_instance
exact ConcreteCategory.bijective_of_isIso
(X.presheaf.stalkSpecializes
(specializes_of_eq (genericPoint_eq_of_dominant j)) ≫ j.stalkMap (genericPoint Y))
noncomputable def functionFieldOpenEquiv
{X Y : Scheme} [IsIntegral X] [IsIntegral Y]
(j : Y ⟶ X) [IsOpenImmersion j] [IsDominant j] :
X.functionField ≃+* Y.functionField :=
RingEquiv.ofBijective (functionFieldPullback j) (functionFieldPullback_bijective_of_open j)
noncomputable def chartFunctionFieldEquiv
(A M : Type u) [CommRing A] [IsDomain A] [Field M]
[Algebra A M] [IsFractionRing A M]
{X : Scheme.{u}} [IsIntegral X]
(j : Spec (.of A) ⟶ X) [IsOpenImmersion j] [IsDominant j] :
M ≃+* X.functionField := by
letI : Algebra A (Spec (.of A)).functionField :=
AlgebraicGeometry.instAlgebraCarrierFunctionFieldSpec (.of A)
letI : IsFractionRing A (Spec (.of A)).functionField :=
functionField_isFractionRing_of_affine (.of A)
exact (IsLocalization.algEquiv (nonZeroDivisors A) M
(Spec (.of A)).functionField).toRingEquiv.trans (functionFieldOpenEquiv j).symm
lemma chartFunctionFieldEquiv_algebraMap
(A M : Type u) [CommRing A] [IsDomain A] [Field M]
[Algebra A M] [IsFractionRing A M]
{X : Scheme.{u}} [IsIntegral X]
(j : Spec (.of A) ⟶ X) [IsOpenImmersion j] [IsDominant j] (a : A) :
functionFieldPullback j (chartFunctionFieldEquiv A M j (algebraMap A M a)) =
@algebraMap A (Spec (.of A)).functionField _ _
(AlgebraicGeometry.instAlgebraCarrierFunctionFieldSpec (.of A)) a := by
let : Algebra A (Spec (.of A)).functionField :=
AlgebraicGeometry.instAlgebraCarrierFunctionFieldSpec (.of A)
let : IsFractionRing A (Spec (.of A)).functionField :=
functionField_isFractionRing_of_affine (.of A)
change (functionFieldOpenEquiv j)
((functionFieldOpenEquiv j).symm
((IsLocalization.algEquiv (nonZeroDivisors A) M
(Spec (.of A)).functionField) (algebraMap A M a))) = _
rw [RingEquiv.apply_symm_apply, AlgEquiv.commutes]
lemma functionFieldPullback_comp
{X Y Z : Scheme} [IsIntegral X] [IsIntegral Y] [IsIntegral Z]
(f : X ⟶ Y) (g : Y ⟶ Z) [IsDominant f] [IsDominant g] :
functionFieldPullback (f ≫ g) = (functionFieldPullback f).comp (functionFieldPullback g) := by
change (Z.presheaf.stalkSpecializes _ ≫ (f ≫ g).stalkMap _).hom =
((Z.presheaf.stalkSpecializes _ ≫ g.stalkMap _) ≫
(Y.presheaf.stalkSpecializes _ ≫ f.stalkMap _)).hom
congr 1
rw [Scheme.Hom.stalkMap_comp, Category.assoc, ← Category.assoc (g.stalkMap _), ← g.stalkSpecializes_stalkMap]
simp only [← Category.assoc, TopCat.Presheaf.stalkSpecializes_comp]
simp only [Category.assoc]
rfl
lemma functionFieldPullback_constant
(K : Type u) [Field K] {X Y : Scheme.{u}} [IsIntegral X] [IsIntegral Y]
(f : X ⟶ Spec (.of K)) (j : Y ⟶ X) [IsDominant j] (c : K) :
functionFieldPullback j (constantToFunctionField K f c) =
constantToFunctionField K (j ≫ f) c := by
unfold constantToFunctionField
simp only [RingHom.comp_apply]
rw [functionFieldPullback_germ]
change Y.germToFunctionField ⊤ (j.appTop (((Scheme.ΓSpecIso (.of K)).inv ≫ f.appTop) c)) = _
rw [Scheme.Hom.comp_appTop]
rfl
lemma constantToFunctionField_spec
(K A : Type u) [Field K] [CommRing A] [IsDomain A] [Algebra K A] (c : K) :
constantToFunctionField K (Spec.map (CommRingCat.ofHom (algebraMap K A))) c =
@algebraMap A (Spec (.of A)).functionField _ _
(AlgebraicGeometry.instAlgebraCarrierFunctionFieldSpec (.of A))
(algebraMap K A c) := by
unfold constantToFunctionField
rw [← Scheme.ΓSpecIso_inv_naturality]
rfl
lemma functionFieldPullback_spec_algebraMap
(A B : Type u) [CommRing A] [IsDomain A] [CommRing B] [IsDomain B]
(h : A →+* B) [IsDominant (Spec.map (CommRingCat.ofHom h))] (a : A) :
functionFieldPullback (Spec.map (CommRingCat.ofHom h))
(@algebraMap A (Spec (.of A)).functionField _ _
(AlgebraicGeometry.instAlgebraCarrierFunctionFieldSpec (.of A)) a) =
@algebraMap B (Spec (.of B)).functionField _ _
(AlgebraicGeometry.instAlgebraCarrierFunctionFieldSpec (.of B)) (h a) := by
change functionFieldPullback (Spec.map (CommRingCat.ofHom h))
((Spec (.of A)).germToFunctionField ⊤ ((Scheme.ΓSpecIso (.of A)).inv a)) = _
rw [functionFieldPullback_germ]
change (Spec (.of B)).germToFunctionField ⊤
(((Scheme.ΓSpecIso (.of A)).inv ≫ (Spec.map (CommRingCat.ofHom h)).appTop) a) = _
rw [← Scheme.ΓSpecIso_inv_naturality]
rfl
end RelativeDenominators
end
end OAI
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