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7f5fab8 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 | 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
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