OModulePresheafOfModules.lean

def ofModules : OModulePresheaf π where
  obj U := Γ(M, U)
  addCommGroup _ := inferInstance
  module U := Scheme.TwoAffineOpenCover.moduleSectionsOfHom π M U
  moduleSections _ := inferInstance
  isScalarTower U := Scheme.TwoAffineOpenCover.isScalarTower_sections π M U
  res h := Scheme.TwoAffineOpenCover.restrictSections π M h
  res_smul h a x := Scheme.Modules.map_smul M (homOfLE h) a x
  res_refl U := by
    let := Scheme.TwoAffineOpenCover.moduleSectionsOfHom π M U
    refine LinearMap.ext fun x => ?_
    change M.presheaf.map (homOfLE (le_refl U)).op x = x
    rw [Subsingleton.elim (homOfLE (le_refl U)).op (𝟙 (op U)), M.presheaf.map_id]
    rfl
  res_comp {U U' U''} h h' := by
    let := Scheme.TwoAffineOpenCover.moduleSectionsOfHom π M U
    let := Scheme.TwoAffineOpenCover.moduleSectionsOfHom π M U''
    refine LinearMap.ext fun x => ?_
    change M.presheaf.map (homOfLE (h.trans h')).op x
      = M.presheaf.map (homOfLE h).op (M.presheaf.map (homOfLE h').op x)
    rw [← CategoryTheory.ConcreteCategory.comp_apply, ← Functor.map_comp]
    rfl

theorem ofModules_obj (U : V.Opens) : (ofModules π M).obj U = Γ(M, U)

theorem ofModules_res_apply {U U' : V.Opens} (h : U ≤ U') (x : Γ(M, U')) :
    (ofModules π M).res h (show (ofModules π M).obj U' from x) = M.presheaf.map (homOfLE h).op x :=
  rfl

theorem ofModules_smul (U : V.Opens) (a : Γ(V, U)) (x : Γ(M, U)) :
    (show (ofModules π M).obj U from a • x) = a • (show (ofModules π M).obj U from x) := rfl

theorem ofModules_algebraMap_smul (U : V.Opens) (r : R) (x : (ofModules π M).obj U) :
    r • x = (Scheme.TwoAffineOpenCover.algebraOfHom π U).algebraMap r • x

theorem ofModules_twoChartSections (𝒱 : V.TwoAffineOpenCover) :
    (ofModules π M).twoChartSections 𝒱 = 𝒱.sectionsOf π M