| import Mathlib |
| import OAI.AlgebraicGeometry.Seshadri.Cohomology.CurveH0Finite |
| import OAI.AlgebraicGeometry.Seshadri.Sheaves.PlaneModules |
|
|
| namespace OAI |
|
|
|
|
| |
| section |
|
|
| namespace MaximalSeshadri.Geometry.BaseSections |
| noncomputable section |
| open AlgebraicGeometry CategoryTheory TopologicalSpace |
|
|
| variable {K : Type} [Field K] {X : Scheme.{0}} |
|
|
| abbrev chartRes (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) |
|
|
| lemma chartRes_comp (k : K →+* Γ(X,⊤)) (M : X.Modules) {U V W : X.Opens} |
| (h : W ≤ V) (g : V ≤ U) |
| (m : Sections (U.ι.appTop.hom.comp k) (M.restrict U.ι) ⊤) : |
| chartRes k M h (chartRes k M g m) = chartRes k M (h.trans g) m := by |
| apply (chartTop k M W).injective |
| rw [chartTop_nested,chartTop_nested,chartTop_nested,res_comp] |
|
|
| lemma chartRes_injective [IsIntegral X] (k : K →+* Γ(X,⊤)) |
| (L : LineBundle X) {U W : X.Opens} (h : W ≤ U) [Nonempty W] : |
| Function.Injective (chartRes k L.sheaf h) := by |
| intro x y hxy |
| apply (chartTop k L.sheaf U).injective |
| apply L.restriction_injective (homOfLE h) |
| change res k L.sheaf h (chartTop k L.sheaf U x) = res k L.sheaf h (chartTop k L.sheaf U y) |
| rw [← chartTop_nested,← chartTop_nested,hxy] |
|
|
| lemma chartRes_range_le (k : K →+* Γ(X,⊤)) (M : X.Modules) {U V W : X.Opens} |
| (h : W ≤ V) (g : V ≤ U) : |
| (chartRes k M (h.trans g)).range ≤ (chartRes k M h).range := by |
| rintro _ ⟨m,rfl⟩ |
| exact ⟨chartRes k M g m,chartRes_comp k M h g m⟩ |
|
|
| theorem chartRes_cross_clearing (k : K →+* Γ(X,⊤)) (M : X.Modules) [M.IsQuasicoherent] |
| (U V W : X.Opens) [IsAffine U.toScheme] (hU : W ≤ U) (hV : W ≤ V) |
| (a : Γ(U.toScheme,⊤)) |
| (ha : (X.homOfLE (show U ⊓ V ≤ U from inf_le_left)).opensRange = U.toScheme.basicOpen a) |
| (m : Sections (W.ι.appTop.hom.comp k) (M.restrict W.ι) ⊤) |
| (hm : m ∈ (chartRes k M hV).range) : |
| ∃ n : ℕ, (X.homOfLE hU).appTop a ^ n • m ∈ (chartRes k M hU).range := by |
| obtain ⟨v,rfl⟩ := hm |
| let P := U ⊓ V |
| have hP : W ≤ P := le_inf hU hV |
| let f := X.homOfLE (show P ≤ U from inf_le_left) |
| let g := X.homOfLE (show P ≤ V from inf_le_right) |
| let w := chartRes k M (show P ≤ V from inf_le_right) v |
| obtain ⟨n,u,hu⟩ := chartMap_clearing (U.ι.appTop.hom.comp k) (P.ι.appTop.hom.comp k) |
| f (nestedBase k inf_le_left) (M.restrict U.ι) (nestedRestriction M inf_le_left) a ha w |
| refine ⟨n,u,?_⟩ |
| have he := congrArg (chartRes k M hP) hu |
| rw [chartRes_comp,chartMap_smul,map_pow,chartRes_comp] at he |
| have har : (X.homOfLE hP).appTop (f.appTop a) = (X.homOfLE hU).appTop a := by |
| rw [← CommRingCat.comp_apply,← Scheme.Hom.comp_appTop] |
| rw [show X.homOfLE hP ≫ f = X.homOfLE hU from Scheme.homOfLE_homOfLE X hP inf_le_left] |
| simpa only [har] using he |
|
|
| end |
| end MaximalSeshadri.Geometry.BaseSections |
|
|
| end |
|
|
|
|
|
|
| end OAI |
|
|
