File size: 5,691 Bytes
a72760f | 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 | import Mathlib.RingTheory.Localization.LocalizationLocalization
import OAI.NumberTheory.PiExponent.Geometry.CurveFunctionFieldCompatibility
namespace OAI
noncomputable section
namespace PiExponent.CurveNormalizationModel
open CategoryTheory AlgebraicGeometry
universe u
variable (A E : Type u) [CommRing A] [IsDomain A] [Field E]
[Algebra A E] [IsFractionRing A E]
theorem isFractionRing_of_intermediate_domain
(B : Type u) [CommRing B] [IsDomain B] [Algebra A B]
[Algebra B E] [IsScalarTower A B E]
(hB : Function.Injective (algebraMap B E)) : IsFractionRing B E where
map_units b := isUnit_iff_ne_zero.mpr (by
intro h
exact (mem_nonZeroDivisors_iff_ne_zero.mp b.2) (hB (by simpa using h)))
surj z := by
obtain ⟨⟨a, s⟩, hs⟩ := IsLocalization.surj (nonZeroDivisors A) z
have hn : algebraMap A B s ≠ 0 := by
intro he
have he' : algebraMap A E s = 0 := by
rw [IsScalarTower.algebraMap_apply A B E, he, map_zero]
exact (mem_nonZeroDivisors_iff_ne_zero.mp s.2)
((IsFractionRing.injective A E) (by simp at he'))
refine ⟨⟨algebraMap A B a, ⟨algebraMap A B s, mem_nonZeroDivisors_iff_ne_zero.mpr hn⟩⟩, ?_⟩
simpa only [← IsScalarTower.algebraMap_apply A B E] using hs
exists_of_eq {a b} h := ⟨1, by simpa using hB h⟩
def affineFunctionFieldEquiv : (Spec (.of A)).functionField ≃+* E := 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) (Spec (.of A)).functionField E).toRingEquiv
@[simp] theorem affineFunctionFieldEquiv_toStalk (a : A) :
affineFunctionFieldEquiv A E (StructureSheaf.toStalk A
(genericPoint (Spec (.of A))) a) = algebraMap A E 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)
exact (IsLocalization.algEquiv (nonZeroDivisors A)
(Spec (.of A)).functionField E).commutes a
theorem affineFunctionFieldEquiv_stalk (q : PrimeSpectrum A)
[Algebra (Localization.AtPrime q.asIdeal) E]
[IsScalarTower A (Localization.AtPrime q.asIdeal) E]
(a : (Spec (.of A)).presheaf.stalk q) :
affineFunctionFieldEquiv A E
((Spec (.of A)).presheaf.stalkSpecializes (genericPoint_specializes q) a) =
algebraMap (Localization.AtPrime q.asIdeal) E ((Spec.stalkIso (.of A) q).hom a) := by
let : Algebra A ((Spec (.of A)).presheaf.stalk q) := StructureSheaf.stalkAlgebra A q
let : IsLocalization.AtPrime ((Spec (.of A)).presheaf.stalk q) q.asIdeal :=
StructureSheaf.IsLocalization.to_stalk A q
have heq : (affineFunctionFieldEquiv A E).toRingHom.comp
((Spec (.of A)).presheaf.stalkSpecializes (genericPoint_specializes q)).hom =
(algebraMap (Localization.AtPrime q.asIdeal) E).comp
(Spec.stalkIso (.of A) q).hom.hom := by
apply IsLocalization.ringHom_ext q.asIdeal.primeCompl
apply RingHom.ext
intro b
change affineFunctionFieldEquiv A E
((Spec (.of A)).presheaf.stalkSpecializes (genericPoint_specializes q)
(StructureSheaf.toStalk A q b)) =
algebraMap (Localization.AtPrime q.asIdeal) E
((Spec.stalkIso (.of A) q).hom (StructureSheaf.toStalk A q b))
have hs : (Spec (.of A)).presheaf.stalkSpecializes (genericPoint_specializes q)
(StructureSheaf.toStalk A q b) =
StructureSheaf.toStalk A (genericPoint (Spec (.of A))) b := by
exact congrArg (fun h => h b)
(StructureSheaf.toStalk_stalkSpecializes (R := A) (genericPoint_specializes q))
have ht : (Spec.stalkIso (.of A) q).hom (StructureSheaf.toStalk A q b) =
algebraMap A (Localization.AtPrime q.asIdeal) b :=
(StructureSheaf.stalkIso A q).symm.commutes b
rw [hs, ht, affineFunctionFieldEquiv_toStalk]
exact IsScalarTower.algebraMap_apply A (Localization.AtPrime q.asIdeal) E b
exact RingHom.congr_fun heq a
theorem affineFunctionFieldEquiv_naturality
(B : Type u) [CommRing B] [IsDomain B] [Algebra A B]
[Algebra B E] [IsFractionRing B E] [IsScalarTower A B E]
[IsOpenImmersion (Spec.map (CommRingCat.ofHom (algebraMap A B)))] :
(affineFunctionFieldEquiv B E).toRingHom.comp
(functionFieldRestriction (Spec.map (CommRingCat.ofHom (algebraMap A B)))).hom =
(affineFunctionFieldEquiv A E).toRingHom := 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)
apply IsLocalization.ringHom_ext (nonZeroDivisors A)
apply RingHom.ext
intro a
change affineFunctionFieldEquiv B E
(functionFieldRestriction (Spec.map (CommRingCat.ofHom (algebraMap A B)))
(StructureSheaf.toStalk A (genericPoint (Spec (.of A))) a)) =
affineFunctionFieldEquiv A E
(StructureSheaf.toStalk A (genericPoint (Spec (.of A))) a)
have h := congrArg (fun h => h a)
(functionFieldRestriction_toStalk (CommRingCat.ofHom (algebraMap A B)))
change functionFieldRestriction (Spec.map (CommRingCat.ofHom (algebraMap A B)))
(StructureSheaf.toStalk A (genericPoint (Spec (.of A))) a) =
StructureSheaf.toStalk B (genericPoint (Spec (.of B))) (algebraMap A B a) at h
rw [h, affineFunctionFieldEquiv_toStalk, affineFunctionFieldEquiv_toStalk]
exact (IsScalarTower.algebraMap_apply A B E a).symm
end PiExponent.CurveNormalizationModel
end
end OAI
|