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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 | import OAI.NumberTheory.PiExponent.Approximation.OpenBaseChange
namespace OAI
namespace PiExponentSeshadri.ClosedPullbackUnit
noncomputable section
open AlgebraicGeometry CategoryTheory TopologicalSpace Opposite
variable {X Y : Scheme.{0}} (f : Y ⟶ X)
def opensEquivalence [IsIso f.base] : X.Opens ≌ Y.Opens :=
Opens.mapMapIso (asIso f.base)
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)
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
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)
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)
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)
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
end OAI
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