openAI-math-dataset / lean /OAI /NumberTheory /PiExponent /Approximation /ClosedPushforwardCoherent.lean
| import OAI.NumberTheory.PiExponent.Approximation.ClosedPushforwardAffine | |
| import OAI.NumberTheory.PiExponent.Approximation.OpenBaseChange | |
| import OAI.NumberTheory.PiExponent.LocalAlgebra.FiniteGlobalPresentation | |
| namespace OAI | |
| noncomputable section | |
| namespace PiExponentSeshadri.ClosedPushforward | |
| open AlgebraicGeometry CategoryTheory CategoryTheory.Limits TopologicalSpace Opposite | |
| variable {X Y : Scheme.{0}} | |
| lemma sections_empty (M : X.Modules) (U : X.Opens) (hU : U = ⊥) : | |
| Subsingleton (M.val.obj (op U)) := by | |
| exact AddCommGrpCat.subsingleton_of_isZero | |
| (TopCat.Sheaf.isTerminalOfEqEmpty ((SheafOfModules.toSheaf.{0} X.ringCatSheaf).obj M) hU).isZero | |
| lemma map_epi (f : X ⟶ Y) [IsClosedImmersion f] {M N : X.Modules} | |
| (φ : M ⟶ N) [Epi φ] : Epi ((Scheme.Modules.pushforward f).map φ) := by | |
| let FX := SheafOfModules.toSheaf.{0} X.ringCatSheaf | |
| let FY := SheafOfModules.toSheaf.{0} Y.ringCatSheaf | |
| let S := ShortComplex.mk _ _ (kernel.condition φ) | |
| have hS : S.ShortExact := { exact := ShortComplex.exact_kernel φ } | |
| have hepi : Epi (FX.map φ) := (ModuleSheafExact.shortExact_map X.ringCatSheaf hS).epi_g | |
| have hφ : TopCat.Presheaf.IsLocallySurjective (FX.map φ).hom := | |
| (TopCat.Sheaf.isLocallySurjective_iff_epi _).mpr hepi | |
| have hloc : TopCat.Presheaf.IsLocallySurjective | |
| (FY.map ((Scheme.Modules.pushforward f).map φ)).hom := by | |
| apply (TopCat.Presheaf.isLocallySurjective_iff _).mpr | |
| intro U t y hy | |
| by_cases him : y ∈ Set.range f | |
| · obtain ⟨x, rfl⟩ := him | |
| obtain ⟨W,hWU,⟨s,hs⟩,hxW⟩ := | |
| (TopCat.Presheaf.isLocallySurjective_iff _).mp hφ (f ⁻¹ᵁ U) t x hy | |
| obtain ⟨V',hV',hpre⟩ := f.isClosedEmbedding.isEmbedding.isInducing.isOpen_iff.mp W.isOpen | |
| let V : Y.Opens := ⟨V', hV'⟩ ⊓ U | |
| have hV : f ⁻¹ᵁ V = W := by | |
| ext z | |
| change (f z ∈ V' ∧ f z ∈ U) ↔ z ∈ W | |
| have hh : f z ∈ V' ↔ z ∈ W := Set.ext_iff.mp hpre z | |
| exact ⟨fun h => hh.mp h.1, fun h => ⟨hh.mpr h,hWU h⟩⟩ | |
| refine ⟨V,inf_le_right,?_,?_,⟩ | |
| · change ∃ s, (φ.val.app (op (f ⁻¹ᵁ V))) s = | |
| N.val.map ((Opens.map f.base).map (homOfLE inf_le_right)).op t | |
| subst W | |
| exact ⟨s,hs⟩ | |
| · exact ⟨(Set.ext_iff.mp hpre x).mpr hxW,hy⟩ | |
| · let V : Y.Opens := U ⊓ ⟨(Set.range f)ᶜ, f.isClosedEmbedding.isClosed_range.isOpen_compl⟩ | |
| have hV : f ⁻¹ᵁ V = ⊥ := by | |
| ext x | |
| change (f x ∈ U ∧ f x ∉ Set.range f) ↔ False | |
| simp | |
| have := sections_empty N (f ⁻¹ᵁ V) hV | |
| refine ⟨V,inf_le_left,⟨0,?_⟩,⟨hy,him⟩⟩ | |
| exact @Subsingleton.elim (N.val.obj (op (f ⁻¹ᵁ V))) (sections_empty N _ hV) _ _ | |
| let : Epi (FY.map ((Scheme.Modules.pushforward f).map φ)) := | |
| (TopCat.Sheaf.isLocallySurjective_iff_epi _).mp hloc | |
| exact FY.epi_of_epi_map inferInstance | |
| instance preservesEpis (f : X ⟶ Y) [IsClosedImmersion f] : | |
| (Scheme.Modules.pushforward f).PreservesEpimorphisms where | |
| preserves {_ _} φ _ := map_epi f φ | |
| instance preservesHomology (f : X ⟶ Y) [IsClosedImmersion f] : | |
| (Scheme.Modules.pushforward f).PreservesHomology := | |
| Functor.preservesHomology_of_preservesEpis_and_kernels _ | |
| instance preservesFiniteColimits (f : X ⟶ Y) [IsClosedImmersion f] : | |
| PreservesFiniteColimits (Scheme.Modules.pushforward f) := | |
| Functor.preservesFiniteColimits_of_preservesHomology _ | |
| open PiExponent.CoherentAffineFinite PiExponent.FiniteGlobalPresentation | |
| open PiExponent.ClosedPushforwardAffine | |
| private theorem presentation_transport {Z : Scheme.{0}} | |
| {M N : SheafOfModules Z.ringCatSheaf} (e : M ≅ N) | |
| (P : M.Presentation) (hP : P.IsFinite) : ∃ Q : N.Presentation, Q.IsFinite := by | |
| let : IsIso e.hom := e.isIso_hom | |
| exact ⟨P.ofIsIso e.hom, | |
| ⟨⟨hP.isFiniteType_generators.finite⟩, ⟨hP.isFiniteType_relations.finite⟩⟩⟩ | |
| instance pushforward_isFinitePresentation (f : X ⟶ Y) [IsClosedImmersion f] | |
| [IsLocallyNoetherian Y] (M : X.Modules) [M.IsFinitePresentation] : | |
| ((Scheme.Modules.pushforward f).obj M).IsFinitePresentation := by | |
| apply isFinitePresentation_of_affine_presentations | |
| intro U | |
| have : IsAffine U.1.toScheme := U.2 | |
| have : IsAffine (f ⁻¹ᵁ U.1).toScheme := U.2.preimage f | |
| let : M.IsQuasicoherent := | |
| (SheafOfModules.IsFinitePresentation.exists_quasicoherentData M).choose.isQuasicoherent | |
| have hM := (locallyFinitelyGenerated_of_finitePresentation M).restrict (f ⁻¹ᵁ U.1).ι | |
| obtain ⟨P,hP⟩ := affine_pushforward_exists_finitePresentation (f ∣_ U.1) | |
| (M.restrict (f ⁻¹ᵁ U.1).ι) hM | |
| exact presentation_transport (Z := U.1.toScheme) (OpenBaseChange.iso f U.1 M).symm P hP | |
| end PiExponentSeshadri.ClosedPushforward | |
| end | |
| end OAI | |