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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 | 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
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