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7f5fab8 | 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 | import Mathlib
import OAI.AlgebraicGeometry.Seshadri.Sheaves.ChartModuleMap
namespace OAI
section
noncomputable section
section
namespace MaximalSeshadri.Geometry.BaseSections
noncomputable section
open AlgebraicGeometry CategoryTheory TopologicalSpace Opposite
variable {K : Type} [CommRing K] {X : Scheme.{0}}
lemma congr_apply (k : K →+* Γ(X,⊤)) (M : X.Modules) {U V : X.Opens}
(h : U = V) (m : Sections k M U) :
congr k M h m = M.presheaf.map (eqToHom h.symm).op m := by
subst V
change m = M.presheaf.map (𝟙 (op U)) m
exact congrArg (fun f : M.presheaf.obj (op U) ⟶ M.presheaf.obj (op U) => f m)
(M.presheaf.map_id (op U)).symm
def nestedRestriction (M : X.Modules) {U W : X.Opens} (h : W ≤ U) :
(M.restrict U.ι).restrict (X.homOfLE h) ≅ M.restrict W.ι :=
((Scheme.Modules.restrictFunctorComp (X.homOfLE h) U.ι).app M).symm ≪≫
(Scheme.Modules.restrictFunctorCongr (X.homOfLE_ι h)).app M
lemma nestedBase (k : K →+* Γ(X,⊤)) {U W : X.Opens} (h : W ≤ U) :
(X.homOfLE h).appTop.hom.comp (U.ι.appTop.hom.comp k) = W.ι.appTop.hom.comp k := by
rw [← RingHom.comp_assoc, ← CommRingCat.hom_comp, ← Scheme.Hom.comp_appTop,
Scheme.homOfLE_ι]
lemma chartTop_nested (k : K →+* Γ(X,⊤)) (M : X.Modules)
{U W : X.Opens} (h : W ≤ U)
(m : Sections (U.ι.appTop.hom.comp k) (M.restrict U.ι) ⊤) :
chartTop k M W (chartMap (U.ι.appTop.hom.comp k) (W.ι.appTop.hom.comp k)
(X.homOfLE h) (nestedBase k h) (M.restrict U.ι) (nestedRestriction M h) m) =
res k M h (chartTop k M U m) := by
simp only [chartTop, LinearEquiv.trans_apply, restrictEquiv, congr_apply,
chartMap, chartModuleMap, LinearMap.coe_comp,
chartModuleEquiv, moduleIsoSections, openImmersionTop, restrictedOpenSections,
nestedRestriction, Iso.trans_hom, Iso.symm_hom, Iso.app_hom]
change (M.presheaf.map _ ≫ M.presheaf.map _ ≫ M.presheaf.map _ ≫
M.presheaf.map _) m = (M.presheaf.map _ ≫ M.presheaf.map _) m
simp only [← M.presheaf.map_comp]
apply congrArg (fun morphism :
M.presheaf.obj (op (U.ι ''ᵁ (⊤ : U.toScheme.Opens))) ⟶ M.presheaf.obj (op W) =>
morphism m)
exact (M.presheaf.map_comp _ _).symm.trans
((congrArg M.presheaf.map (Subsingleton.elim _ _)).trans
(M.presheaf.map_comp _ _))
lemma chartMap_range_coherent (k : K →+* Γ(X,⊤)) (M : X.Modules)
{U W : X.Opens} (h : W ≤ U) :
(chartMap (U.ι.appTop.hom.comp k) (W.ι.appTop.hom.comp k) (X.homOfLE h)
(nestedBase k h) (M.restrict U.ι) (nestedRestriction M h)).range.map
(chartTop k M W).toLinearMap = (res k M h).range := by
ext x
constructor
· rintro ⟨y,⟨z,rfl⟩,rfl⟩
exact ⟨chartTop k M U z, (chartTop_nested k M h z).symm⟩
· rintro ⟨z,rfl⟩
refine ⟨_, ⟨(chartTop k M U).symm z,rfl⟩, ?_⟩
exact (chartTop_nested k M h _).trans
(congrArg (res k M h) ((chartTop k M U).apply_symm_apply z))
end
end MaximalSeshadri.Geometry.BaseSections
end
end
end
end OAI
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