openAI-math-dataset / lean /OAI /NumberTheory /PiExponent /Approximation /ClosedPushforwardLinear.lean
| import OAI.NumberTheory.PiExponent.Approximation.ClosedImmersionSerreTransfer | |
| import OAI.NumberTheory.PiExponent.Cohomology.EulerExact | |
| namespace OAI | |
| namespace PiExponent.ClosedImmersionSerreTransfer | |
| noncomputable section | |
| open AlgebraicGeometry CategoryTheory CategoryTheory.Limits CategoryTheory.Abelian | |
| open TopologicalSpace Opposite PiExponentSeshadri.Geometry | |
| variable {X Y : Scheme.{0}} (f : X ⟶ Y) | |
| local instance closedPushforwardLinearSourceHasExt : HasExt.{1} X.Modules := schemeHasExt | |
| local instance closedPushforwardLinearTargetHasExt : HasExt.{1} Y.Modules := schemeHasExt | |
| lemma baseScalars_comp_actual (p : Y ⟶ Spec (CommRingCat.of ℂ)) : | |
| baseScalars (f ≫ p) = f.appTop.hom.comp (baseScalars p) := by | |
| ext c | |
| simp only [baseScalars, Scheme.Hom.comp_appTop, CommRingCat.hom_comp, | |
| RingHom.comp_apply] | |
| lemma scalar_naturality (p : Y ⟶ Spec (CommRingCat.of ℂ)) (U : Y.Opens) (r : ℂ) : | |
| restrictScalar X (f ⁻¹ᵁ U) (baseScalars (f ≫ p) r) = | |
| f.app U (restrictScalar Y U (baseScalars p r)) := by | |
| rw [baseScalars_comp_actual] | |
| change X.presheaf.map (homOfLE le_top).op (f.appTop (baseScalars p r)) = _ | |
| exact (CategoryTheory.congr_fun (f.naturality (homOfLE le_top).op) (baseScalars p r)).symm | |
| theorem complexLinearPushforward (p : Y ⟶ Spec (CommRingCat.of ℂ)) : | |
| letI := sheafComplexLinear (f ≫ p) | |
| letI := sheafComplexLinear p | |
| (Scheme.Modules.pushforward f).Linear ℂ := by | |
| let := sheafComplexLinear (f ≫ p) | |
| let := sheafComplexLinear p | |
| constructor | |
| intro M N g r | |
| ext U m | |
| let y : Γ(N, f ⁻¹ᵁ U) := g.val.app (op (f ⁻¹ᵁ U)) m | |
| change restrictScalar X (f ⁻¹ᵁ U) (baseScalars (f ≫ p) r) • y = | |
| f.app U (restrictScalar Y U (baseScalars p r)) • y | |
| rw [scalar_naturality] | |
| variable [IsClosedImmersion f] | |
| lemma cohomologyMap_comp {M N : X.Modules} {n m k : ℕ} | |
| (x : cohomology M n) (y : Ext.{1} M N m) (h : n + m = k) : | |
| cohomologyMap f N k (x.comp y h) = | |
| (cohomologyMap f M n x).comp (y.mapExactFunctor (Scheme.Modules.pushforward f)) h := | |
| PiExponentSeshadri.ExtSectionComparison.mapFrom_comp | |
| (Scheme.Modules.pushforward f) (structureMap f) x y h | |
| def typedCohomologyMap (M : X.Modules) (n : ℕ) : | |
| cohomology M n →+ cohomology ((Scheme.Modules.pushforward f).obj M) n := | |
| cohomologyMap f M n | |
| lemma typedCohomologyMap_comp {M N : X.Modules} {n m k : ℕ} | |
| (x : cohomology M n) (y : Ext.{1} M N m) (h : n + m = k) : | |
| typedCohomologyMap f N k (x.comp y h) = | |
| (typedCohomologyMap f M n x).comp | |
| (y.mapExactFunctor (Scheme.Modules.pushforward f)) h := | |
| cohomologyMap_comp f x y h | |
| lemma typedCohomologyMap_bijective (M : X.Modules) (n : ℕ) : | |
| Function.Bijective (typedCohomologyMap f M n) := | |
| cohomologyMap_bijective f M n | |
| def cohomologyLinearMap (p : Y ⟶ Spec (CommRingCat.of ℂ)) (M : X.Modules) (n : ℕ) : | |
| letI := Module.compHom (cohomology M n) (baseScalars (f ≫ p)) | |
| letI := Module.compHom (cohomology ((Scheme.Modules.pushforward f).obj M) n) (baseScalars p) | |
| cohomology M n →ₗ[ℂ] cohomology ((Scheme.Modules.pushforward f).obj M) n := by | |
| letI := Module.compHom (cohomology M n) (baseScalars (f ≫ p)) | |
| letI := Module.compHom (cohomology ((Scheme.Modules.pushforward f).obj M) n) (baseScalars p) | |
| exact { toFun := typedCohomologyMap f M n | |
| map_add' := (typedCohomologyMap f M n).map_add | |
| map_smul' := by | |
| intro r x | |
| change typedCohomologyMap f M n ((baseScalars (f ≫ p) r) • x) = | |
| (baseScalars p r) • (typedCohomologyMap f M n x) | |
| rw [Ext.smul_eq_comp_mk₀, typedCohomologyMap_comp, Ext.smul_eq_comp_mk₀, | |
| Ext.mapExactFunctor_mk₀] | |
| congr 2 | |
| ext U m | |
| change Γ(M, f ⁻¹ᵁ U) at m | |
| change restrictScalar X (f ⁻¹ᵁ U) (baseScalars (f ≫ p) r) • m = | |
| f.app U (restrictScalar Y U (baseScalars p r)) • m | |
| rw [scalar_naturality] } | |
| lemma cohomologyLinearMap_eq (p : Y ⟶ Spec (CommRingCat.of ℂ)) (M : X.Modules) (n : ℕ) : | |
| ⇑(cohomologyLinearMap f p M n) = typedCohomologyMap f M n := rfl | |
| def cohomologyLinearEquiv (p : Y ⟶ Spec (CommRingCat.of ℂ)) (M : X.Modules) (n : ℕ) : | |
| letI := Module.compHom (cohomology M n) (baseScalars (f ≫ p)) | |
| letI := Module.compHom (cohomology ((Scheme.Modules.pushforward f).obj M) n) (baseScalars p) | |
| cohomology M n ≃ₗ[ℂ] cohomology ((Scheme.Modules.pushforward f).obj M) n := by | |
| exact LinearEquiv.ofBijective (cohomologyLinearMap f p M n) | |
| (by rw [cohomologyLinearMap_eq]; exact typedCohomologyMap_bijective f M n) | |
| theorem finiteDimensional_of_pushforward | |
| (p : Y ⟶ Spec (CommRingCat.of ℂ)) (M : X.Modules) (n : ℕ) | |
| (hfinite : letI := Module.compHom (cohomology ((Scheme.Modules.pushforward f).obj M) n) (baseScalars p) | |
| FiniteDimensional ℂ (cohomology ((Scheme.Modules.pushforward f).obj M) n)) : | |
| letI := Module.compHom (cohomology M n) (baseScalars (f ≫ p)) | |
| FiniteDimensional ℂ (cohomology M n) := by | |
| let := Module.compHom (cohomology M n) (baseScalars (f ≫ p)) | |
| let := Module.compHom (cohomology ((Scheme.Modules.pushforward f).obj M) n) (baseScalars p) | |
| have := hfinite | |
| exact FiniteDimensional.of_injective (cohomologyLinearEquiv f p M n).toLinearMap | |
| (cohomologyLinearEquiv f p M n).injective | |
| lemma cohomologyDimension_pushforward | |
| (p : Y ⟶ Spec (CommRingCat.of ℂ)) (M : X.Modules) (n : ℕ) : | |
| cohomologyDimension p ((Scheme.Modules.pushforward f).obj M) n = | |
| cohomologyDimension (f ≫ p) M n := by | |
| let := Module.compHom (cohomology M n) (baseScalars (f ≫ p)) | |
| let := Module.compHom (cohomology ((Scheme.Modules.pushforward f).obj M) n) (baseScalars p) | |
| exact (cohomologyLinearEquiv f p M n).finrank_eq.symm | |
| theorem euler_pushforward (p : Y ⟶ Spec (CommRingCat.of ℂ)) (M : X.Modules) (d : ℕ) : | |
| eulerCharacteristic p d ((Scheme.Modules.pushforward f).obj M) = | |
| eulerCharacteristic (f ≫ p) d M := by | |
| apply Finset.sum_congr rfl | |
| intro n hn | |
| rw [cohomologyDimension_pushforward] | |
| end | |
| end PiExponent.ClosedImmersionSerreTransfer | |
| end OAI | |