openAI-math-dataset / lean /OAI /NumberTheory /PiExponent /Approximation /ClosedImmersionModules.lean
| import OAI.NumberTheory.PiExponent.Approximation.ClosedPullbackUnit | |
| namespace OAI | |
| namespace PiExponentSeshadri.ClosedImmersionModules | |
| noncomputable section | |
| open AlgebraicGeometry CategoryTheory TopologicalSpace Opposite | |
| variable {X Y : Scheme.{0}} (f : Y ⟶ X) | |
| def additiveFullyFaithful [IsClosedImmersion f] : | |
| ((Functor.whiskeringLeft _ _ AddCommGrpCat).obj (Opens.map f.base).op).FullyFaithful := by | |
| let h := f.isClosedEmbedding.isEmbedding.isInducing | |
| let adj := h.adjunction | |
| have hc (V : Y.Opens) : IsIso (adj.counit.app V) := by | |
| have he : adj.counit.app V = eqToHom (h.map_functorObj V) := Subsingleton.elim _ _ | |
| exact he.symm ▸ (eqToIso (h.map_functorObj V)).isIso_hom | |
| let : IsIso adj.counit := NatIso.isIso_of_isIso_app _ | |
| let := adj.fullyFaithfulROfIsIsoCounit.full | |
| let := adj.fullyFaithfulROfIsIsoCounit.faithful | |
| let W := (MorphismProperty.isomorphisms Y.Opens).inverseImage (Opens.map f.base) | |
| let : (Opens.map f.base).IsLocalization W := adj.isLocalization | |
| let : (Opens.map f.base).op.IsLocalization W.op := | |
| (Functor.IsLocalization.op_iff (Opens.map f.base) W).mpr inferInstance | |
| exact Localization.fullyFaithfulWhiskeringLeft (Opens.map f.base).op W.op AddCommGrpCat | |
| lemma preimage_surjective [IsClosedImmersion f] : | |
| Function.Surjective (fun U : X.Opens => f ⁻¹ᵁ U) := by | |
| intro V | |
| let h := f.isClosedEmbedding.isEmbedding.isInducing | |
| exact ⟨h.functorObj V, h.map_functorObj V⟩ | |
| def additivePreimage [IsClosedImmersion f] {M N : Y.Modules} | |
| (a : (Scheme.Modules.pushforward f).obj M ⟶ (Scheme.Modules.pushforward f).obj N) : | |
| M.presheaf ⟶ N.presheaf := | |
| (additiveFullyFaithful f).preimage a.mapPresheaf | |
| lemma additivePreimage_app [IsClosedImmersion f] {M N : Y.Modules} | |
| (a : (Scheme.Modules.pushforward f).obj M ⟶ (Scheme.Modules.pushforward f).obj N) | |
| (U : X.Opens) : (additivePreimage f a).app (op (f ⁻¹ᵁ U)) = a.app U := by | |
| exact congrArg (fun b : (Opens.map f.base).op ⋙ M.presheaf ⟶ | |
| (Opens.map f.base).op ⋙ N.presheaf => b.app (op U)) | |
| ((additiveFullyFaithful f).map_preimage (X := M.presheaf) (Y := N.presheaf) a.mapPresheaf) | |
| lemma additivePreimage_smul [IsClosedImmersion f] | |
| {M N : Y.Modules} | |
| (a : (Scheme.Modules.pushforward f).obj M ⟶ (Scheme.Modules.pushforward f).obj N) | |
| (V : Y.Opens) (r : Γ(Y,V)) (m : Γ(M,V)) : | |
| (additivePreimage f a).app (op V) (r • m) = | |
| r • (additivePreimage f a).app (op V) m := by | |
| obtain ⟨U, rfl⟩ := preimage_surjective f V | |
| apply N.isSheaf.section_ext | |
| intro y hy | |
| obtain ⟨W, hW, hyW, hWU⟩ := exists_isAffineOpen_mem_and_subset (X := X) (x := f y) (U := U) hy | |
| let j : W ⟶ U := homOfLE hWU | |
| let k : f ⁻¹ᵁ W ⟶ f ⁻¹ᵁ U := (Opens.map f.base).map j | |
| refine ⟨f ⁻¹ᵁ W, leOfHom k, hyW, ?_⟩ | |
| have hn (z : Γ(M,f ⁻¹ᵁ U)) : | |
| (additivePreimage f a).app (op (f ⁻¹ᵁ W)) (M.presheaf.map k.op z) = | |
| N.presheaf.map k.op ((additivePreimage f a).app (op (f ⁻¹ᵁ U)) z) := | |
| CategoryTheory.congr_fun ((additivePreimage f a).naturality k.op) z | |
| have hs : N.presheaf.map k.op (r • (additivePreimage f a).app (op (f ⁻¹ᵁ U)) m) = | |
| Y.presheaf.map k.op r • N.presheaf.map k.op ((additivePreimage f a).app (op (f ⁻¹ᵁ U)) m) := | |
| N.val.map_smul k.op r ((additivePreimage f a).app (op (f ⁻¹ᵁ U)) m) | |
| change N.presheaf.map k.op ((additivePreimage f a).app (op (f ⁻¹ᵁ U)) (r • m)) = | |
| N.presheaf.map k.op (r • (additivePreimage f a).app (op (f ⁻¹ᵁ U)) m) | |
| rw [← hn, hs, ← hn, additivePreimage_app] | |
| obtain ⟨c, hc⟩ := f.app_surjective W hW (Y.presheaf.map k.op r) | |
| have hm := M.val.map_smul k.op r m | |
| change M.presheaf.map k.op (r • m) = Y.presheaf.map k.op r • M.presheaf.map k.op m at hm | |
| rw [hm, ← hc] | |
| exact a.app_smul (r := c) (x := M.presheaf.map k.op m) | |
| def modulePreimage [IsClosedImmersion f] {M N : Y.Modules} | |
| (a : (Scheme.Modules.pushforward f).obj M ⟶ (Scheme.Modules.pushforward f).obj N) : | |
| M ⟶ N where | |
| val.app V := ModuleCat.ofHom (X := M.val.obj V) (Y := N.val.obj V) | |
| { toAddHom := ((additivePreimage f a).app V).hom.toAddHom | |
| map_smul' := fun r m => additivePreimage_smul f a V.unop r m } | |
| val.naturality {V W} j := by | |
| ext m | |
| exact CategoryTheory.congr_fun ((additivePreimage f a).naturality j) m | |
| def pushforwardFullyFaithful [IsClosedImmersion f] : | |
| (Scheme.Modules.pushforward f).FullyFaithful where | |
| preimage a := modulePreimage f a | |
| map_preimage a := by | |
| ext U m | |
| exact CategoryTheory.congr_fun (additivePreimage_app f a U) m | |
| preimage_map a := by | |
| apply Scheme.Modules.hom_ext | |
| intro V | |
| obtain ⟨U, rfl⟩ := preimage_surjective f V | |
| exact additivePreimage_app f ((Scheme.Modules.pushforward f).map a) U | |
| theorem counit_isIso [IsClosedImmersion f] : | |
| IsIso (Scheme.Modules.pullbackPushforwardAdjunction f).counit := by | |
| let := (pushforwardFullyFaithful f).full | |
| let := (pushforwardFullyFaithful f).faithful | |
| exact (Scheme.Modules.pullbackPushforwardAdjunction f).counit_isIso_of_R_fully_faithful | |
| theorem unit_isIso_of_pushforward [IsClosedImmersion f] | |
| {M : X.Modules} {N : Y.Modules} | |
| (e : M ≅ (Scheme.Modules.pushforward f).obj N) : | |
| IsIso ((Scheme.Modules.pullbackPushforwardAdjunction f).unit.app M) := by | |
| let : IsIso (Scheme.Modules.pullbackPushforwardAdjunction f).counit := counit_isIso f | |
| let := (Scheme.Modules.pullbackPushforwardAdjunction f).fullyFaithfulROfIsIsoCounit.full | |
| let := (Scheme.Modules.pullbackPushforwardAdjunction f).fullyFaithfulROfIsIsoCounit.faithful | |
| exact (Scheme.Modules.pullbackPushforwardAdjunction f).isIso_unit_app_of_iso e | |
| end | |
| end PiExponentSeshadri.ClosedImmersionModules | |
| end OAI | |