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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 | 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
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