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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 | 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
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