| import Mathlib |
| import OAI.AlgebraicGeometry.Seshadri.Divisors.PolynomialSections |
| import OAI.AlgebraicGeometry.Seshadri.Sheaves.ChartModuleMap |
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| namespace OAI |
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| section |
| noncomputable section |
| |
| section |
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| namespace MaximalSeshadri.Geometry.BaseSections |
| noncomputable section |
| open AlgebraicGeometry CategoryTheory TopologicalSpace |
| open scoped Polynomial |
|
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| variable {K : Type} [CommRing K] {X Y : Scheme.{0}} |
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| lemma eval_transport (kX : K →+* Γ(X,⊤)) (kY : K →+* Γ(Y,⊤)) |
| (f : Y ⟶ X) (hk : f.appTop.hom.comp kX = kY) (t : Γ(X,⊤)) |
| (p : K[X]) : f.appTop ((Polynomial.eval₂RingHom kX t) p) = |
| (Polynomial.eval₂RingHom kY (f.appTop t)) p := by |
| have h : f.appTop.hom.comp (Polynomial.eval₂RingHom kX t) = |
| Polynomial.eval₂RingHom kY (f.appTop t) := by |
| apply Polynomial.ringHom_ext |
| · intro r |
| simpa using RingHom.congr_fun hk r |
| · simp |
| exact RingHom.congr_fun h p |
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| def chartPolynomialMap (kX : K →+* Γ(X,⊤)) (kY : K →+* Γ(Y,⊤)) |
| (f : Y ⟶ X) [IsOpenImmersion f] (hk : f.appTop.hom.comp kX = kY) |
| (M : X.Modules) {N : Y.Modules} (e : M.restrict f ≅ N) (t : Γ(X,⊤)) : |
| letI := polynomialModule kX t M ⊤ |
| letI := polynomialModule kY (f.appTop t) N ⊤ |
| Sections kX M ⊤ →ₗ[K[X]] Sections kY N ⊤ := by |
| letI := polynomialModule kX t M ⊤ |
| letI := polynomialModule kY (f.appTop t) N ⊤ |
| refine { toFun := chartMap kX kY f hk M e |
| map_add' := (chartMap kX kY f hk M e).map_add |
| map_smul' := ?_ } |
| intro p m |
| change chartMap kX kY f hk M e ((Polynomial.eval₂RingHom kX t) p • m) = |
| (Polynomial.eval₂RingHom kY (f.appTop t)) p • chartMap kX kY f hk M e m |
| rw [chartMap_smul, eval_transport kX kY f hk] |
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| theorem chartPolynomialMap_localize [IsAffine X] |
| (kX : K →+* Γ(X,⊤)) (kY : K →+* Γ(Y,⊤)) |
| (f : Y ⟶ X) [IsOpenImmersion f] (hk : f.appTop.hom.comp kX = kY) |
| (M : X.Modules) [M.IsQuasicoherent] {N : Y.Modules} (e : M.restrict f ≅ N) |
| (t : Γ(X,⊤)) (ht : f.opensRange = X.basicOpen t) : |
| letI := polynomialModule kX t M ⊤ |
| letI := polynomialModule kY (f.appTop t) N ⊤ |
| IsLocalizedModule.Away (Polynomial.X : K[X]) |
| (chartPolynomialMap kX kY f hk M e t) := by |
| let : Algebra K[X] Γ(X,⊤) := (Polynomial.eval₂RingHom kX t).toAlgebra |
| let := polynomialModule kX t M ⊤ |
| let := polynomialModule kY (f.appTop t) N ⊤ |
| let : Module Γ(X,⊤) (Sections kY N ⊤) := |
| Module.compHom (OpenSections N ⊤) f.appTop.hom |
| let : IsScalarTower K[X] Γ(X,⊤) (Sections kX M ⊤) := |
| .of_algebraMap_smul (fun _ _ => rfl) |
| let : IsScalarTower K[X] Γ(X,⊤) (Sections kY N ⊤) := |
| .of_algebraMap_smul (fun p m => by |
| change f.appTop ((Polynomial.eval₂RingHom kX t) p) • (m : OpenSections N ⊤) = |
| (Polynomial.eval₂RingHom kY (f.appTop t)) p • (m : OpenSections N ⊤) |
| rw [eval_transport kX kY f hk]) |
| let F : Sections kX M ⊤ →ₗ[Γ(X,⊤)] Sections kY N ⊤ := chartModuleMap M f e |
| have he : algebraMap K[X] Γ(X,⊤) Polynomial.X = t := by |
| change (Polynomial.eval₂RingHom kX t) Polynomial.X = t |
| simp |
| let : IsLocalizedModule (.powers (algebraMap K[X] Γ(X,⊤) Polynomial.X)) F := by |
| rw [he] |
| exact chartModuleMap_localize M f e t ht |
| have h := IsLocalizedModule.restrictScalars_powers (Polynomial.X : K[X]) F |
| exact h |
| end |
| end MaximalSeshadri.Geometry.BaseSections |
| end |
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| end |
| end |
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| end OAI |
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