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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 | import Mathlib
import OAI.AlgebraicGeometry.Seshadri.Cohomology.LaurentCech
import OAI.AlgebraicGeometry.Seshadri.Geometry.ChartPolynomial
namespace OAI
section
noncomputable section
section
namespace MaximalSeshadri.Geometry.BaseSections
noncomputable section
open AlgebraicGeometry CategoryTheory TopologicalSpace
open scoped Polynomial
variable {K : Type} [Field K] {U V W : Scheme.{0}} [IsAffine U] [IsAffine V]
theorem two_chart_cokernel_finite
(kU : K →+* Γ(U,⊤)) (kV : K →+* Γ(V,⊤)) (kW : K →+* Γ(W,⊤))
(a : W ⟶ U) (b : W ⟶ V) [IsOpenImmersion a] [IsOpenImmersion b]
(hka : a.appTop.hom.comp kU = kW) (hkb : b.appTop.hom.comp kV = kW)
(u : Γ(U,⊤)) (v : Γ(V,⊤))
(ha : a.opensRange = U.basicOpen u) (hb : b.opensRange = V.basicOpen v)
(huv : a.appTop u * b.appTop v = 1)
(hu : (Polynomial.eval₂RingHom kU u).Finite)
(L : LineBundle U) (P : LineBundle V) (N : W.Modules)
(e : L.sheaf.restrict a ≅ N) (d : P.sheaf.restrict b ≅ N) :
Module.Finite K (Sections kW N ⊤ ⧸
((chartMap kU kW a hka L.sheaf e).range ⊔
(chartMap kV kW b hkb P.sheaf d).range)) := by
let := polynomialModule kU u L.sheaf ⊤
let := polynomialModule kV v P.sheaf ⊤
let := polynomialModule kW (a.appTop u) N ⊤
have := polynomialTower kU u L.sheaf ⊤
have := polynomialTower kV v P.sheaf ⊤
have := polynomialTower kW (a.appTop u) N ⊤
have : Module.Finite K[X] (Sections kU L.sheaf ⊤) := polynomial_finite kU u L hu
let F := chartPolynomialMap kU kW a hka L.sheaf e u
let G := chartMap kV kW b hkb P.sheaf d
have : IsLocalizedModule.Away (Polynomial.X : K[X]) F :=
chartPolynomialMap_localize kU kW a hka L.sheaf e u ha
have hInv (y : Sections kV P.sheaf ⊤) :
(Polynomial.X : K[X]) • G ((Polynomial.X : K[X]) • y) = G y := by
change (Polynomial.eval₂RingHom kW (a.appTop u)) Polynomial.X •
G ((Polynomial.eval₂RingHom kV v) Polynomial.X • y) = G y
simp only [Polynomial.coe_eval₂RingHom, Polynomial.eval₂_X]
rw [chartMap_smul, ← mul_smul, huv, one_smul]
have hDen (x : OpenSections N ⊤) : ∃ n : ℕ, ∃ y : Sections kV P.sheaf ⊤,
(Polynomial.X : K[X])^n • G y = x := by
let : Module Γ(V,⊤) (OpenSections N ⊤) :=
Module.compHom (OpenSections N ⊤) b.appTop.hom
let G' := chartModuleMap P.sheaf b d
have : IsLocalizedModule.Away v G' := chartModuleMap_localize P.sheaf b d v hb
obtain ⟨n,y,hy⟩ := IsLocalizedModule.Away.surj G' v x
refine ⟨n,y,?_⟩
change (Polynomial.eval₂RingHom kW (a.appTop u)) (Polynomial.X^n) • G y = x
simp only [map_pow, Polynomial.coe_eval₂RingHom, Polynomial.eval₂_X]
change b.appTop (v^n) • (x : OpenSections N ⊤) = G' y at hy
change (a.appTop u)^n • (G' y : OpenSections N ⊤) = x
rw [← hy, map_pow, ← mul_smul, ← mul_pow, huv, one_pow, one_smul]
exact LaurentCech.cokernel_finite F G hInv hDen
end
end MaximalSeshadri.Geometry.BaseSections
end
end
end
end OAI
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