| import Mathlib |
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| namespace OAI |
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| noncomputable section |
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| namespace CampanaPeternellSix.Cotangent.Affine |
| open CategoryTheory AlgebraicGeometry Opposite TopologicalSpace |
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| universe u |
| variable (R S : CommRingCat.{u}) (φ : R ⟶ S) |
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| |
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| def specGlobalPushforwardIso : |
| Scheme.Modules.pushforward (Spec.map φ) ⋙ moduleSpecΓFunctor ≅ |
| moduleSpecΓFunctor ⋙ ModuleCat.restrictScalars φ.hom := by |
| refine NatIso.ofComponents (fun M => ?_) ?_ |
| · refine LinearEquiv.toModuleIso |
| (X₁ := moduleSpecΓFunctor.obj ((Scheme.Modules.pushforward (Spec.map φ)).obj M)) |
| (X₂ := (ModuleCat.restrictScalars φ.hom).obj (moduleSpecΓFunctor.obj M)) |
| (m₁ := ModuleCat.isModule (moduleSpecΓFunctor.obj |
| ((Scheme.Modules.pushforward (Spec.map φ)).obj M))) |
| (m₂ := ModuleCat.isModule ((ModuleCat.restrictScalars φ.hom).obj |
| (moduleSpecΓFunctor.obj M))) |
| { toFun := fun x => x |
| invFun := fun x => x |
| left_inv := fun _ => rfl |
| right_inv := fun _ => rfl |
| map_add' := fun _ _ => rfl |
| map_smul' := ?_ } |
| intro r m |
| change Γ(M, ⊤) at m |
| change (Spec.map φ).appTop ((Scheme.ΓSpecIso R).inv r) • m = |
| (Scheme.ΓSpecIso S).inv (φ r) • m |
| exact congrArg (fun a => a • m) |
| (CategoryTheory.congr_fun (Scheme.ΓSpecIso_inv_naturality φ).symm r) |
| · intro M N f |
| ext x |
| rfl |
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| |
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| def tildePullbackIso : |
| tilde.functor R ⋙ Scheme.Modules.pullback (Spec.map φ) ≅ |
| ModuleCat.extendScalars φ.hom ⋙ tilde.functor S := |
| ((tilde.adjunction.comp (Scheme.Modules.pullbackPushforwardAdjunction (Spec.map φ))).ofNatIsoRight (specGlobalPushforwardIso R S φ)).leftAdjointUniq |
| ((ModuleCat.extendRestrictScalarsAdj φ.hom).comp tilde.adjunction) |
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| |
| def affinePullbackAdjunction := |
| (tilde.adjunction.comp (Scheme.Modules.pullbackPushforwardAdjunction (Spec.map φ))).ofNatIsoRight |
| (specGlobalPushforwardIso R S φ) |
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| lemma tildePullbackIso_homEquiv (M : ModuleCat R) (N : ModuleCat S) |
| (g : (ModuleCat.extendScalars φ.hom).obj M ⟶ N) : |
| (affinePullbackAdjunction R S φ).homEquiv M (tilde N) |
| ((tildePullbackIso R S φ).hom.app M ≫ tilde.map g) = |
| ((ModuleCat.extendRestrictScalarsAdj φ.hom).comp tilde.adjunction).homEquiv |
| M (tilde N) (tilde.map g) := by |
| erw [Adjunction.homEquiv_naturality_right] |
| change (affinePullbackAdjunction R S φ).homEquiv _ _ |
| (((affinePullbackAdjunction R S φ).leftAdjointUniq |
| ((ModuleCat.extendRestrictScalarsAdj φ.hom).comp tilde.adjunction)).hom.app M) ≫ _ = _ |
| erw [Adjunction.homEquiv_leftAdjointUniq_hom_app, ← Adjunction.homEquiv_unit] |
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| lemma tilde_homEquiv_apply (M : ModuleCat R) (N : (Spec R).Modules) |
| (f : tilde M ⟶ N) (m : M) : |
| tilde.adjunction.homEquiv M N f m = f.val.app (op ⊤) (tilde.toOpen M ⊤ m) := by |
| erw [Adjunction.homEquiv_unit] |
| rfl |
|
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| lemma affinePullback_homEquiv_apply (M : ModuleCat R) (N : (Spec S).Modules) |
| (f : (Scheme.Modules.pullback (Spec.map φ)).obj (tilde M) ⟶ N) (m : M) : |
| (affinePullbackAdjunction R S φ).homEquiv M N f m = |
| ((Scheme.Modules.pullbackPushforwardAdjunction (Spec.map φ)).homEquiv _ _ f).val.app |
| (op ⊤) (tilde.toOpen M ⊤ m) := by |
| erw [affinePullbackAdjunction, Adjunction.homEquiv_ofNatIsoRight_apply, |
| Adjunction.comp_homEquiv] |
| change tilde.adjunction.homEquiv M _ |
| ((Scheme.Modules.pullbackPushforwardAdjunction (Spec.map φ)).homEquiv _ _ f) m = _ |
| exact tilde_homEquiv_apply R _ _ _ _ |
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| lemma extendTilde_homEquiv_apply (M : ModuleCat R) (N : ModuleCat S) |
| (g : (ModuleCat.extendScalars φ.hom).obj M ⟶ N) (m : M) : |
| ((ModuleCat.extendRestrictScalarsAdj φ.hom).comp tilde.adjunction).homEquiv |
| M (tilde N) (tilde.map g) m = |
| tilde.toOpen N ⊤ (g (ModuleCat.extendRestrictScalarsAdj φ.hom |>.unit.app M m)) := by |
| rw [Adjunction.comp_homEquiv] |
| change (ModuleCat.extendRestrictScalarsAdj φ.hom).homEquiv _ _ |
| (tilde.adjunction.homEquiv _ _ (tilde.map g)) m = _ |
| erw [ModuleCat.extendRestrictScalarsAdj_homEquiv_apply, tilde_homEquiv_apply] |
| exact CategoryTheory.congr_fun (tilde.toOpen_map_app g ⊤) _ |
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| end CampanaPeternellSix.Cotangent.Affine |
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| namespace CampanaPeternellSix.AffineBaseChange |
| open CategoryTheory AlgebraicGeometry Opposite |
| universe u |
| variable (R S : CommRingCat.{u}) (φ : R ⟶ S) |
| |
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| def sectionMap (E : (Spec R).Modules) : |
| moduleSpecΓFunctor.obj E →ₛₗ[φ.hom] |
| moduleSpecΓFunctor.obj ((Scheme.Modules.pullback (Spec.map φ)).obj E) where |
| toFun m := ((Scheme.Modules.pullbackPushforwardAdjunction (Spec.map φ)).unit.app E).val.app |
| (op ⊤) m |
| map_add' := map_add _ |
| map_smul' r m := by |
| change Γ(E, ⊤) at m |
| change ((Scheme.Modules.pullbackPushforwardAdjunction (Spec.map φ)).unit.app E).val.app |
| (op ⊤) (((Scheme.ΓSpecIso R).inv r) • m) = _ |
| let η : E ⟶ (Scheme.Modules.pushforward (Spec.map φ)).obj |
| ((Scheme.Modules.pullback (Spec.map φ)).obj E) := |
| (Scheme.Modules.pullbackPushforwardAdjunction (Spec.map φ)).unit.app E |
| have hη := Scheme.Modules.Hom.app_smul η ((Scheme.ΓSpecIso R).inv r) m |
| erw [hη] |
| let n : Γ((Scheme.Modules.pullback (Spec.map φ)).obj E, ⊤) := |
| ((Scheme.Modules.pullbackPushforwardAdjunction (Spec.map φ)).unit.app E).val.app (op ⊤) m |
| change ((Spec.map φ).appTop ((Scheme.ΓSpecIso R).inv r)) • n = |
| ((Scheme.ΓSpecIso S).inv (φ r)) • n |
| exact congrArg (fun a => a • n) |
| (CategoryTheory.congr_fun (Scheme.ΓSpecIso_inv_naturality φ).symm r) |
|
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| open Cotangent.Affine |
|
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| def globalPullbackIso (E : (Spec R).Modules) [E.IsQuasicoherent] : |
| (ModuleCat.extendScalars φ.hom).obj (moduleSpecΓFunctor.obj E) ≅ |
| moduleSpecΓFunctor.obj ((Scheme.Modules.pullback (Spec.map φ)).obj E) := by |
| let iso : tilde (moduleSpecΓFunctor.obj E) ≅ E := |
| @asIso _ _ _ _ E.fromTildeΓ (Scheme.Modules.isIso_fromTildeΓ_of_isQuasicoherent E) |
| exact tilde.toTildeΓNatIso.app _ ≪≫ |
| moduleSpecΓFunctor.mapIso ((tildePullbackIso R S φ).app _).symm ≪≫ |
| moduleSpecΓFunctor.mapIso |
| ((Scheme.Modules.pullback (Spec.map φ)).mapIso iso) |
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| lemma tilde_unit_comparison (M : ModuleCat R) (m : M) : |
| ((tildePullbackIso R S φ).hom.app M).val.app (op ⊤) |
| (sectionMap R S φ (tilde M) (tilde.toOpen M ⊤ m)) = |
| tilde.toOpen ((ModuleCat.extendScalars φ.hom).obj M) ⊤ |
| ((ModuleCat.extendRestrictScalarsAdj φ.hom).unit.app M m) := by |
| have h := congrArg (fun f => f m) |
| (tildePullbackIso_homEquiv R S φ M ((ModuleCat.extendScalars φ.hom).obj M) |
| (𝟙 ((ModuleCat.extendScalars φ.hom).obj M))) |
| erw [affinePullback_homEquiv_apply R S φ M |
| (tilde ((ModuleCat.extendScalars φ.hom).obj M)) |
| ((tildePullbackIso R S φ).hom.app M ≫ |
| tilde.map (𝟙 ((ModuleCat.extendScalars φ.hom).obj M))) m, |
| extendTilde_homEquiv_apply R S φ M ((ModuleCat.extendScalars φ.hom).obj M) |
| (𝟙 ((ModuleCat.extendScalars φ.hom).obj M)) m] at h |
| have h_id : tilde.map (𝟙 ((ModuleCat.extendScalars φ.hom).obj M)) = 𝟙 _ := |
| (tilde.functor S).map_id _ |
| erw [h_id] at h |
| erw [Category.comp_id, Adjunction.homEquiv_unit] at h |
| exact h |
|
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| lemma sectionMap_naturality {E F : (Spec R).Modules} (f : E ⟶ F) |
| (m : moduleSpecΓFunctor.obj E) : |
| (moduleSpecΓFunctor.map ((Scheme.Modules.pullback (Spec.map φ)).map f)) |
| (sectionMap R S φ E m) = |
| sectionMap R S φ F (moduleSpecΓFunctor.map f m) := by |
| have h := (Scheme.Modules.pullbackPushforwardAdjunction (Spec.map φ)).unit.naturality f |
| exact congrArg (fun g => g.val.app (op ⊤) m) h.symm |
|
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| lemma counit_top (E : (Spec R).Modules) (m : moduleSpecΓFunctor.obj E) : |
| moduleSpecΓFunctor.map E.fromTildeΓ |
| (tilde.toOpen (moduleSpecΓFunctor.obj E) ⊤ m) = m := by |
| have h := tilde.adjunction.right_triangle_components E |
| exact congrArg (fun g => g m) h |
|
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| lemma globalPullbackIso_one (E : (Spec R).Modules) [E.IsQuasicoherent] |
| (m : moduleSpecΓFunctor.obj E) : |
| (globalPullbackIso R S φ E).hom |
| ((ModuleCat.extendRestrictScalarsAdj φ.hom).unit.app (moduleSpecΓFunctor.obj E) m) = |
| sectionMap R S φ E m := by |
| change moduleSpecΓFunctor.map ((Scheme.Modules.pullback (Spec.map φ)).map E.fromTildeΓ) |
| (((tildePullbackIso R S φ).inv.app (moduleSpecΓFunctor.obj E)).val.app (op ⊤) |
| (tilde.toOpen ((ModuleCat.extendScalars φ.hom).obj (moduleSpecΓFunctor.obj E)) ⊤ |
| ((ModuleCat.extendRestrictScalarsAdj φ.hom).unit.app (moduleSpecΓFunctor.obj E) m))) = _ |
| rw [← tilde_unit_comparison R S φ] |
| have h : ((tildePullbackIso R S φ).inv.app (moduleSpecΓFunctor.obj E)).val.app (op ⊤) |
| (((tildePullbackIso R S φ).hom.app (moduleSpecΓFunctor.obj E)).val.app (op ⊤) |
| (sectionMap R S φ (tilde (moduleSpecΓFunctor.obj E)) |
| (tilde.toOpen (moduleSpecΓFunctor.obj E) ⊤ m))) = |
| sectionMap R S φ (tilde (moduleSpecΓFunctor.obj E)) |
| (tilde.toOpen (moduleSpecΓFunctor.obj E) ⊤ m) := by |
| exact congrArg (fun f => f.val.app (op ⊤) |
| (sectionMap R S φ (tilde (moduleSpecΓFunctor.obj E)) |
| (tilde.toOpen (moduleSpecΓFunctor.obj E) ⊤ m))) |
| ((tildePullbackIso R S φ).app (moduleSpecΓFunctor.obj E)).hom_inv_id |
| rw [h] |
| exact (sectionMap_naturality R S φ E.fromTildeΓ |
| (tilde.toOpen (moduleSpecΓFunctor.obj E) ⊤ m)).trans |
| (congrArg (sectionMap R S φ E) (counit_top R E m)) |
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| |
| theorem isBaseChange_sections (E : (Spec R).Modules) [E.IsQuasicoherent] : |
| letI : Algebra R S := φ.hom.toAlgebra |
| let N := moduleSpecΓFunctor.obj ((Scheme.Modules.pullback (Spec.map φ)).obj E) |
| letI : Module R N := Module.compHom N φ.hom |
| letI : IsScalarTower R S N := IsScalarTower.of_algebraMap_smul (fun _ _ => rfl) |
| IsBaseChange S |
| ({ toFun := sectionMap R S φ E |
| map_add' := map_add _ |
| map_smul' := fun r m => (sectionMap R S φ E).map_smul' r m } : |
| moduleSpecΓFunctor.obj E →ₗ[R] N) := by |
| let : Algebra R S := φ.hom.toAlgebra |
| let N := moduleSpecΓFunctor.obj ((Scheme.Modules.pullback (Spec.map φ)).obj E) |
| let : Module R N := Module.compHom N φ.hom |
| let : IsScalarTower R S N := IsScalarTower.of_algebraMap_smul (fun _ _ => rfl) |
| exact IsBaseChange.of_equiv (globalPullbackIso R S φ E).toLinearEquiv |
| (globalPullbackIso_one R S φ E) |
| end CampanaPeternellSix.AffineBaseChange |
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| end |
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| end OAI |
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