| import OAI.NumberTheory.PiExponent.Approximation.OpenBaseChange |
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| namespace OAI |
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| namespace PiExponentSeshadri.ClosedPullbackUnit |
| noncomputable section |
| open AlgebraicGeometry CategoryTheory TopologicalSpace Opposite |
| variable {X Y : Scheme.{0}} (f : Y ⟶ X) |
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| def opensEquivalence [IsIso f.base] : X.Opens ≌ Y.Opens := |
| Opens.mapMapIso (asIso f.base) |
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| lemma preimage_surjective [IsIso f.base] : |
| Function.Surjective (fun U : X.Opens => f ⁻¹ᵁ U) := by |
| intro V |
| refine ⟨(opensEquivalence f).inverse.obj V, ?_⟩ |
| exact le_antisymm (leOfHom ((opensEquivalence f).counitIso.app V).hom) |
| (leOfHom ((opensEquivalence f).counitIso.app V).inv) |
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| def additivePreimage [IsIso f.base] {M N : Y.Modules} |
| (a : (Scheme.Modules.pushforward f).obj M ⟶ (Scheme.Modules.pushforward f).obj N) : |
| M.presheaf ⟶ N.presheaf := by |
| let : (Opens.map f.base).IsEquivalence := (opensEquivalence f).isEquivalence_functor |
| exact ((Functor.whiskeringLeft _ _ AddCommGrpCat).obj (Opens.map f.base).op).preimage a.mapPresheaf |
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| lemma additivePreimage_app [IsIso f.base] {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 |
| let : (Opens.map f.base).IsEquivalence := (opensEquivalence f).isEquivalence_functor |
| exact congrArg (fun b : (Opens.map f.base).op ⋙ M.presheaf ⟶ |
| (Opens.map f.base).op ⋙ N.presheaf => b.app (op U)) |
| (((Functor.whiskeringLeft _ _ AddCommGrpCat).obj (Opens.map f.base).op).map_preimage (X := M.presheaf) (Y := N.presheaf) a.mapPresheaf) |
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| lemma additivePreimage_smul [IsClosedImmersion f] [IsIso f.base] |
| {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) |
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| def modulePreimage [IsClosedImmersion f] [IsIso f.base] {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] [IsIso f.base] : |
| (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] (hf : Function.Surjective f) : |
| IsIso (Scheme.Modules.pullbackPushforwardAdjunction f).counit := by |
| let : IsIso f.base := (TopCat.isIso_iff_isHomeomorph f.base).mpr |
| (isHomeomorph_iff_isEmbedding_surjective.mpr ⟨f.isClosedEmbedding.isEmbedding, hf⟩) |
| 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] (hf : Function.Surjective 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 hf |
| 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 |
|
|
| lemma subscheme_surjective_of_support_top (I : X.IdealSheafData) (hI : I.support = ⊤) : |
| Function.Surjective I.subschemeι := by |
| rw [← Set.range_eq_univ, I.range_subschemeι, hI] |
| rfl |
|
|
| theorem subscheme_counit_isIso (I : X.IdealSheafData) (hI : I.support = ⊤) : |
| IsIso (Scheme.Modules.pullbackPushforwardAdjunction I.subschemeι).counit := |
| counit_isIso I.subschemeι (subscheme_surjective_of_support_top I hI) |
|
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| theorem subscheme_unit_isIso_of_pushforward (I : X.IdealSheafData) (hI : I.support = ⊤) |
| {M : X.Modules} {N : I.subscheme.Modules} |
| (e : M ≅ (Scheme.Modules.pushforward I.subschemeι).obj N) : |
| IsIso ((Scheme.Modules.pullbackPushforwardAdjunction I.subschemeι).unit.app M) := |
| unit_isIso_of_pushforward I.subschemeι (subscheme_surjective_of_support_top I hI) e |
|
|
| end |
| end PiExponentSeshadri.ClosedPullbackUnit |
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| end OAI |
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