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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 | import OAI.NumberTheory.PiExponent.Approximation.FrameSubopens
namespace OAI
namespace PiExponentSeshadri.Frames
noncomputable section
open AlgebraicGeometry CategoryTheory TopologicalSpace Opposite
variable {X : Scheme} {M : X.Modules} {U V W : X.Opens}
lemma restrictOpenFrame_app (h : V ≤ U) (e : M.restrict U.ι ≅ O U.toScheme)
(A : V.toScheme.Opens) (x : Γ(M.restrict V.ι,A)) :
(restrictOpenFrame h e).hom.app A x =
X.presheaf.map (homOfLE (show V.ι ''ᵁ A ≤ U.ι ''ᵁ (X.homOfLE h ''ᵁ A) from by
simp [← Scheme.Hom.comp_image])).op
(e.hom.app (X.homOfLE h ''ᵁ A)
(M.presheaf.map (homOfLE
(show U.ι ''ᵁ (X.homOfLE h ''ᵁ A) ≤ V.ι ''ᵁ A from by
simp [← Scheme.Hom.comp_image])).op x)) := by
simp only [restrictOpenFrame, Iso.trans_hom, Iso.symm_hom,
Scheme.Modules.Hom.comp_app, Iso.app_inv, Iso.app_hom,
Scheme.Modules.restrictFunctorCongr_inv_app_app,
Scheme.Modules.restrictFunctorComp_hom_app_app]
change ((X.homOfLE h).appIso A).hom (e.hom.app (X.homOfLE h ''ᵁ A)
((M.presheaf.map _ ≫ M.presheaf.map _) x)) = _
simp only [Scheme.Hom.appIso_hom', Scheme.homOfLE_appLE]
erw [← Functor.map_comp]
rfl
lemma restrictOpenFrame_trans (h : W ≤ V) (k : V ≤ U)
(e : M.restrict U.ι ≅ O U.toScheme) :
restrictOpenFrame h (restrictOpenFrame k e) = restrictOpenFrame (h.trans k) e := by
apply Iso.ext
ext A x
have hh := restrictOpenFrame_app h (restrictOpenFrame k e) A x
let y := M.presheaf.map (homOfLE
(show V.ι ''ᵁ (X.homOfLE h ''ᵁ A) ≤ W.ι ''ᵁ A from by
simp [← Scheme.Hom.comp_image])).op x
have hk := restrictOpenFrame_app k e (X.homOfLE h ''ᵁ A) y
have hhk := congrArg (X.presheaf.map (homOfLE
(show W.ι ''ᵁ A ≤ V.ι ''ᵁ (X.homOfLE h ''ᵁ A) from by
simp [← Scheme.Hom.comp_image])).op) hk
have hc := restrictOpenFrame_app (h.trans k) e A x
refine hh.trans (hhk.trans (Eq.trans ?_ hc.symm))
change (X.presheaf.map _ ≫ X.presheaf.map _)
(e.hom.app (X.homOfLE k ''ᵁ (X.homOfLE h ''ᵁ A))
((M.presheaf.map _ ≫ M.presheaf.map _) x)) =
X.presheaf.map _ (e.hom.app (X.homOfLE (h.trans k) ''ᵁ A)
(M.presheaf.map _ x))
erw [← Functor.map_comp, ← Functor.map_comp]
let B := X.homOfLE k ''ᵁ (X.homOfLE h ''ᵁ A)
let C := X.homOfLE (h.trans k) ''ᵁ A
have hA : C ≤ B := by
simp only [B, C, ← Scheme.Hom.comp_image, Scheme.homOfLE_homOfLE]
exact le_rfl
have hB : U.ι ''ᵁ B ≤ W.ι ''ᵁ A := by
simp [B, ← Scheme.Hom.comp_image]
have hC : W.ι ''ᵁ A ≤ U.ι ''ᵁ C := by
simp [C, ← Scheme.Hom.comp_image]
have hn := CategoryTheory.congr_fun
(e.hom.mapPresheaf.naturality (homOfLE hA).op)
(M.presheaf.map (homOfLE hB).op x)
change e.hom.app C ((M.restrict U.ι).presheaf.map (homOfLE hA).op
(M.presheaf.map (homOfLE hB).op x)) =
U.toScheme.presheaf.map (homOfLE hA).op
(e.hom.app B (M.presheaf.map (homOfLE hB).op x)) at hn
refine Eq.trans ?_ (Eq.trans (congrArg (X.presheaf.map (homOfLE hC).op) hn).symm ?_)
· change X.presheaf.map _ (e.hom.app B (M.presheaf.map _ x)) =
(X.presheaf.map _ ≫ X.presheaf.map _) (e.hom.app B (M.presheaf.map _ x))
erw [← Functor.map_comp]
rfl
· apply congrArg (X.presheaf.map (homOfLE hC).op)
apply congrArg (e.hom.app C)
change (M.presheaf.map _ ≫ M.presheaf.map _) x = M.presheaf.map _ x
erw [← Functor.map_comp]
rfl
end
end PiExponentSeshadri.Frames
end OAI
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