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12dee50 | 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 | import Mathlib
import OAI.AlgebraicGeometry.Seshadri.Geometry.ChartPolynomial
import OAI.AlgebraicGeometry.Seshadri.Cohomology.LaurentH0
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_intersection_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)
(hv : (Polynomial.eval₂RingHom kV v).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 ↥((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 ⊤
let := polynomialTower kU u L.sheaf ⊤
let := polynomialTower kV v P.sheaf ⊤
let := polynomialTower kW (a.appTop u) N ⊤
let : Module.Finite K[X] (Sections kU L.sheaf ⊤) := polynomial_finite kU u L hu
let : Module.Finite K[X] (Sections kV P.sheaf ⊤) := polynomial_finite kV v P hv
let F := chartPolynomialMap kU kW a hka L.sheaf e u
let G := chartMap kV kW b hkb P.sheaf d
let : 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]
exact LaurentCech.intersection_finite F G hInv
end
end MaximalSeshadri.Geometry.BaseSections
end
end
end
end OAI
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