| import Mathlib |
| import OAI.AlgebraicGeometry.SectionFields.ScalarCharts |
|
|
| namespace OAI |
|
|
| /-! Normality and principal-divisor descent under finite Galois base change. -/ |
|
|
| noncomputable section |
| open AlgebraicGeometry CategoryTheory CategoryTheory.Limits TopologicalSpace Order Polynomial |
| open scoped TensorProduct WithZero |
| universe u |
|
|
| namespace RelativeDenominators |
| section |
| open AlgebraicGeometry CategoryTheory CategoryTheory.Limits |
| open scoped TensorProduct |
|
|
| |
| lemma integrallyClosed_of_affine_open_immersion |
| {X : Scheme.{u}} (hNormal : ∀ x : X, IsIntegrallyClosed (X.presheaf.stalk x)) |
| (A : Type u) [CommRing A] [IsDomain A] |
| (j : Spec (.of A) ⟶ X) [IsOpenImmersion j] : IsIntegrallyClosed A := by |
| apply IsIntegrallyClosed.of_localization_maximal |
| intro p hp hpmax |
| let x : Spec (.of A) := (⟨p, inferInstance⟩ : PrimeSpectrum A) |
| let := hNormal (j x) |
| have hs : IsIntegrallyClosed ((Spec (.of A)).presheaf.stalk x) := |
| IsIntegrallyClosed.of_equiv (asIso (j.stalkMap x)).commRingCatIsoToRingEquiv |
| exact IsIntegrallyClosed.of_equiv (h := hs) |
| (StructureSheaf.stalkIso A (⟨p, inferInstance⟩ : PrimeSpectrum A)).toRingEquiv.symm |
|
|
| lemma integrallyClosed_spec_stalk (A : Type u) [CommRing A] [IsDomain A] |
| [IsIntegrallyClosed A] (x : Spec (.of A)) : |
| IsIntegrallyClosed ((Spec (.of A)).presheaf.stalk x) := by |
| change PrimeSpectrum A at x |
| let : IsIntegrallyClosed (Localization.AtPrime x.asIdeal) := |
| isIntegrallyClosed_of_isLocalization _ x.asIdeal.primeCompl |
| x.asIdeal.primeCompl_le_nonZeroDivisors |
| exact IsIntegrallyClosed.of_equiv (StructureSheaf.stalkIso A x).toRingEquiv |
|
|
| |
| lemma scalarExtensionChart_range |
| (K A L : Type u) [Field K] [CommRing A] [Field L] |
| [Algebra K A] [Algebra K L] |
| {X : Scheme.{u}} (f : X ⟶ Spec (.of K)) |
| (j : Spec (.of A) ⟶ X) |
| (hj : Spec.map (CommRingCat.ofHom (algebraMap K A)) = j ≫ f) : |
| Set.range (scalarExtensionChart K A L f j hj) = |
| pullback.fst f (Spec.map (CommRingCat.ofHom (algebraMap K L))) ⁻¹' Set.range j := by |
| have hs : Set.range (pullbackSpecIso K A L).inv = Set.univ := by |
| simp |
| unfold scalarExtensionChart |
| rw [Scheme.Hom.comp_base, TopCat.coe_comp, Set.range_comp, hs, |
| Set.image_univ, AlgebraicGeometry.Scheme.Pullback.range_map] |
| simp |
| |
| |
| |
| lemma scalarExtensionChart_normal_stalk |
| (K A L : Type u) [Field K] [CommRing A] [IsDomain A] [Field L] |
| [Algebra K A] [Algebra K L] [IsGalois K L] [Module.Finite K L] |
| {X : Scheme.{u}} (f : X ⟶ Spec (.of K)) [GeometricallyIntegral f] |
| (hNormal : ∀ x : X, IsIntegrallyClosed (X.presheaf.stalk x)) |
| (j : Spec (.of A) ⟶ X) [IsOpenImmersion j] |
| (hj : Spec.map (CommRingCat.ofHom (algebraMap K A)) = j ≫ f) |
| (x : Spec (.of (A ⊗[K] L))) : |
| IsIntegrallyClosed ((pullback f (Spec.map (CommRingCat.ofHom (algebraMap K L)))).presheaf.stalk |
| (scalarExtensionChart K A L f j hj x)) := by |
| let := integrallyClosed_of_affine_open_immersion hNormal A j |
| let := scalarExtensionChart_domain K A L f j hj |
| let := tensor_isIntegrallyClosed_of_galois K A L (FractionRing A) |
| let := integrallyClosed_spec_stalk (A ⊗[K] L) x |
| exact IsIntegrallyClosed.of_equiv |
| (asIso ((scalarExtensionChart K A L f j hj).stalkMap x)).commRingCatIsoToRingEquiv.symm |
| |
| |
| |
| theorem galois_baseChange_normal |
| (K L : Type u) [Field K] [Field L] [Algebra K L] [IsGalois K L] |
| [Module.Finite K L] {X : Scheme.{u}} [IsIntegral X] |
| (f : X ⟶ Spec (.of K)) [GeometricallyIntegral f] |
| (hNormal : ∀ x : X, IsIntegrallyClosed (X.presheaf.stalk x)) |
| (y : (pullback f (Spec.map (CommRingCat.ofHom (algebraMap K L))) : Scheme)) : |
| IsIntegrallyClosed ((pullback f |
| (Spec.map (CommRingCat.ofHom (algebraMap K L)))).presheaf.stalk y) := by |
| let z := pullback.fst f (Spec.map (CommRingCat.ofHom (algebraMap K L))) y |
| let i := X.affineOpenCover.idx z |
| let A := X.affineOpenCover.X i |
| let j : Spec A ⟶ X := X.affineOpenCover.f i |
| let : IsOpenImmersion j := X.affineOpenCover.map_prop i |
| have hc : z ∈ Set.range j := X.affineOpenCover.covers z |
| let : Nonempty (Spec A) := ⟨hc.choose⟩ |
| let : IsIntegral (Spec A) := isIntegral_of_isOpenImmersion j |
| have hA : IsDomain A := (affine_isIntegral_iff A).mp inferInstance |
| let := hA |
| let b : CommRingCat.of K ⟶ A := (Spec.map_surjective (j ≫ f)).choose |
| let : Algebra K A := b.hom.toAlgebra |
| have hj : Spec.map (CommRingCat.ofHom (algebraMap K A)) = j ≫ f := |
| (Spec.map_surjective (j ≫ f)).choose_spec |
| have hy : y ∈ Set.range (scalarExtensionChart K A L f j hj) := by |
| rw [scalarExtensionChart_range] |
| exact hc |
| obtain ⟨x, hx⟩ := hy |
| have hn := @scalarExtensionChart_normal_stalk K A L _ A.commRing hA _ _ _ _ _ |
| X f _ hNormal j _ hj x |
| exact hx ▸ hn |
| |
| end |
| |
| attribute [local instance] Units.mulDistribMulActionRight |
| attribute [local instance 2000] CommRingCat.commRing |
| |
| |
| |
| noncomputable def fieldAffineChartRing (X : Scheme.{u}) [IsIntegral X] : CommRingCat.{u} := |
| X.affineOpenCover.X (X.affineOpenCover.idx (genericPoint X)) |
| |
| noncomputable def fieldAffineChartMap (X : Scheme.{u}) [IsIntegral X] : |
| Spec (fieldAffineChartRing X) ⟶ X := |
| X.affineOpenCover.f (X.affineOpenCover.idx (genericPoint X)) |
| |
| instance (X : Scheme.{u}) [IsIntegral X] : IsOpenImmersion (fieldAffineChartMap X) := by |
| exact X.affineOpenCover.map_prop _ |
| |
| instance (X : Scheme.{u}) [IsIntegral X] : Nonempty (Spec (fieldAffineChartRing X)) := by |
| obtain ⟨x, _⟩ := X.affineOpenCover.covers (genericPoint X) |
| exact ⟨x⟩ |
| |
| instance fieldAffineChartRing_isDomain (X : Scheme.{u}) [IsIntegral X] : IsDomain (fieldAffineChartRing X) := by |
| apply (affine_isIntegral_iff _).mp |
| exact isIntegral_of_isOpenImmersion (fieldAffineChartMap X) |
| |
| instance (X : Scheme.{u}) [IsIntegral X] : IsDominant (fieldAffineChartMap X) := |
| isDominant_of_openIntegral _ |
| |
| noncomputable def fieldAffineChartHom (K : Type u) [Field K] |
| {X : Scheme.{u}} [IsIntegral X] (f : X ⟶ Spec (.of K)) : |
| CommRingCat.of K ⟶ fieldAffineChartRing X := |
| (Spec.map_surjective (fieldAffineChartMap X ≫ f)).choose |
| |
| lemma fieldAffineChartHom_spec (K : Type u) [Field K] |
| {X : Scheme.{u}} [IsIntegral X] (f : X ⟶ Spec (.of K)) : |
| Spec.map (fieldAffineChartHom K f) = fieldAffineChartMap X ≫ f := |
| (Spec.map_surjective (fieldAffineChartMap X ≫ f)).choose_spec |
| |
| @[instance_reducible] noncomputable def fieldAffineChartAlgebra (K : Type u) [Field K] |
| {X : Scheme.{u}} [IsIntegral X] (f : X ⟶ Spec (.of K)) : |
| Algebra K (fieldAffineChartRing X) := |
| (fieldAffineChartHom K f).hom.toAlgebra |
| |
| lemma fieldAffineChart_algebraMap_spec (K : Type u) [Field K] |
| {X : Scheme.{u}} [IsIntegral X] (f : X ⟶ Spec (.of K)) : |
| letI := fieldAffineChartAlgebra K f |
| Spec.map (CommRingCat.ofHom (algebraMap K (fieldAffineChartRing X))) = |
| fieldAffineChartMap X ≫ f := |
| fieldAffineChartHom_spec K f |
| |
| |
| @[instance_reducible] noncomputable def schemeBaseChangeGaloisAction |
| (K L : Type u) [Field K] [Field L] [Algebra K L] [Module.Finite K L] |
| {X : Scheme.{u}} [IsIntegral X] |
| (f : X ⟶ Spec (.of K)) [GeometricallyIntegral f] : |
| MulSemiringAction (Gal(L/K)) |
| (pullback f (Spec.map (CommRingCat.ofHom (algebraMap K L)))).functionField := by |
| letI := fieldAffineChartAlgebra K f |
| exact scalarExtensionGaloisAction K (fieldAffineChartRing X) L |
| (FractionRing (fieldAffineChartRing X)) f (fieldAffineChartMap X) |
| (fieldAffineChart_algebraMap_spec K f) |
| |
| |
| |
| theorem same_divisor_descends_from_galois_extension |
| (K L : Type u) [Field K] [Field L] [Algebra K L] [IsGalois K L] |
| [Module.Finite K L] {X : Scheme.{u}} [IsIntegral X] |
| (f : X ⟶ Spec (.of K)) [GeometricallyIntegral f] [IsProper f] |
| (hNormal : ∀ x : X, IsIntegrallyClosed (X.presheaf.stalk x)) |
| (u : (pullback f (Spec.map (CommRingCat.ofHom (algebraMap K L)))).functionFieldˣ) |
| (hdiv : letI := schemeBaseChangeGaloisAction K L f |
| letI : IsNoetherian (pullback f |
| (Spec.map (CommRingCat.ofHom (algebraMap K L)))) := {} |
| ∀ σ : Gal(L/K), principalDivisor _ (σ • u) = principalDivisor _ u) : |
| letI : IsNoetherian (pullback f |
| (Spec.map (CommRingCat.ofHom (algebraMap K L)))) := {} |
| ∃ v : X.functionFieldˣ, |
| principalDivisor _ (Units.map (functionFieldPullback (pullback.fst f |
| (Spec.map (CommRingCat.ofHom (algebraMap K L))))).toMonoidHom v) = |
| principalDivisor _ u := by |
| let := fieldAffineChartAlgebra K f |
| let := fieldAffineChartRing_isDomain X |
| have hmain := @same_divisor_descends_from_galois_baseChange K (fieldAffineChartRing X) L |
| (FractionRing (fieldAffineChartRing X)) _ _ (fieldAffineChartRing_isDomain X) |
| _ _ _ _ _ _ _ _ _ _ X _ f _ _ (fieldAffineChartMap X) _ _ |
| (fieldAffineChart_algebraMap_spec K f) (galois_baseChange_normal K L f hNormal) u |
| exact hmain hdiv |
| |
| end RelativeDenominators |
| end |
| |
| end OAI |
| |
