| import OAI.NumberTheory.PiExponent.Cohomology.EulerExact |
| import OAI.NumberTheory.PiExponent.Cohomology.SerreVanishing |
|
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| namespace OAI |
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| namespace PiExponent.CoherentExtFiniteness |
| noncomputable section |
| open CategoryTheory CategoryTheory.Limits CategoryTheory.Abelian AlgebraicGeometry |
| open PiExponentSeshadri.Geometry |
|
|
| theorem finite_middle_of_exact {K U V W : Type*} [Field K] |
| [AddCommGroup U] [Module K U] [AddCommGroup V] [Module K V] |
| [AddCommGroup W] [Module K W] [Module.Finite K U] [Module.Finite K W] |
| (f : U →ₗ[K] V) (g : V →ₗ[K] W) (h : Function.Exact f g) : |
| Module.Finite K V := by |
| have he : Function.Exact f g.rangeRestrict := by |
| intro x |
| rw [Subtype.ext_iff] |
| exact h x |
| exact Module.Finite.of_exact he g.surjective_rangeRestrict |
|
|
| universe w v u t k |
| variable {K : Type k} [Field K] {C : Type u} [Category.{v} C] [Abelian C] |
| [Linear K C] [HasExt.{w} C] |
|
|
| def extLinearEquivBiproduct (A : C) {J : Type*} [Fintype J] |
| {B : J → C} {c : Bicone B} (hc : c.IsBilimit) (q : ℕ) : |
| Ext.{w} A c.pt q ≃ₗ[K] ∀ j, Ext.{w} A (B j) q := |
| { Ext.addEquivBiproduct A hc q with |
| map_smul' := by |
| intro a x |
| funext j |
| exact Ext.smul_comp x (Ext.mk₀ (c.π j)) (add_zero q) a } |
|
|
| theorem ext_finite_of_biproduct (A : C) {J : Type*} [Fintype J] |
| {B : J → C} {c : Bicone B} (hc : c.IsBilimit) (q : ℕ) |
| (hfinite : ∀ j, Module.Finite K (Ext.{w} A (B j) q)) : |
| Module.Finite K (Ext.{w} A c.pt q) := by |
| let (j : J) : Module.Finite K (Ext.{w} A (B j) q) := hfinite j |
| exact Module.Finite.of_surjective (extLinearEquivBiproduct (K := K) A hc q).symm.toLinearMap |
| (extLinearEquivBiproduct (K := K) A hc q).symm.surjective |
|
|
| structure FiniteCohomologyPresentation {ι : Type t} (A : C) (F : ι → C) (i : ι) where |
| kernel : ι |
| middle : C |
| left : F kernel ⟶ middle |
| right : middle ⟶ F i |
| comp_zero : left ≫ right = 0 |
| shortExact : (ShortComplex.mk left right comp_zero).ShortExact |
| finite : ∀ q, Module.Finite K (Ext.{w} A middle q) |
|
|
| theorem ext_finite_of_presentations {ι : Type t} (A : C) (F : ι → C) (l : ℕ) |
| (hbound : ∀ i q, l ≤ q → ∀ x : Ext.{w} A (F i) q, x = 0) |
| (presentation : ∀ i, FiniteCohomologyPresentation (K := K) A F i) |
| (i : ι) (q : ℕ) : Module.Finite K (Ext.{w} A (F i) q) := by |
| have hdown : ∀ k q, l ≤ q + k → ∀ i, Module.Finite K (Ext.{w} A (F i) q) := by |
| intro k |
| induction k with |
| | zero => |
| intro q hq i |
| let : Subsingleton (Ext.{w} A (F i) q) := |
| ⟨fun x y => (hbound i q (by omega) x).trans (hbound i q (by omega) y).symm⟩ |
| infer_instance |
| | succ k ih => |
| intro q hq i |
| let P := presentation i |
| let S := ShortComplex.mk P.left P.right P.comp_zero |
| let : Module.Finite K (Ext.{w} A S.X₂ q) := P.finite q |
| let : Module.Finite K (Ext.{w} A S.X₁ (q+1)) := ih (q+1) (by omega) P.kernel |
| exact finite_middle_of_exact |
| (PiExponentSeshadri.Cohomology.cohomologyMap₂ (K := K) A (S := S) q) |
| (PiExponentSeshadri.Cohomology.cohomologyBoundary (K := K) A P.shortExact q) |
| (PiExponentSeshadri.Cohomology.cohomology_exact₃ (K := K) A P.shortExact q) |
| exact hdown l q (Nat.le_add_left _ _) i |
|
|
| variable {X : Scheme.{0}} |
|
|
| theorem cohomology_finite_of_biproduct |
| (p : X ⟶ Spec (CommRingCat.of ℂ)) {J : Type*} [Fintype J] |
| {B : J → X.Modules} {c : Bicone B} (hc : c.IsBilimit) (q : ℕ) |
| (hfinite : ∀ j, letI := Module.compHom (cohomology (B j) q) (baseScalars p) |
| FiniteDimensional ℂ (cohomology (B j) q)) : |
| letI := Module.compHom (cohomology c.pt q) (baseScalars p) |
| FiniteDimensional ℂ (cohomology c.pt q) := by |
| let := sheafComplexLinear p |
| have hfin : ∀ j, Module.Finite ℂ (Ext.{1} (structureSheaf X) (B j) q) := by |
| intro j |
| change @Module.Finite ℂ (cohomology (B j) q) _ _ (complexExtModule p (B j) q) |
| rw [complexExtModule_eq] |
| exact hfinite j |
| have hf := ext_finite_of_biproduct (K := ℂ) (structureSheaf X) hc q hfin |
| change @Module.Finite ℂ (cohomology c.pt q) _ _ (complexExtModule p c.pt q) at hf |
| rw [complexExtModule_eq] at hf |
| exact hf |
|
|
| structure CoherentFinitePresentation (p : X ⟶ Spec (CommRingCat.of ℂ)) (M : X.Modules) where |
| kernel : X.Modules |
| kernel_coherent : kernel.IsFinitePresentation |
| middle : X.Modules |
| left : kernel ⟶ middle |
| right : middle ⟶ M |
| comp_zero : left ≫ right = 0 |
| shortExact : (ShortComplex.mk left right comp_zero).ShortExact |
| middle_finite : ∀ q, letI := Module.compHom (cohomology middle q) (baseScalars p) |
| FiniteDimensional ℂ (cohomology middle q) |
|
|
| theorem coherent_cohomology_finite_of_presentations |
| [IsNoetherian X] [IsAffineHom (pullback.diagonal (terminal.from X))] |
| (p : X ⟶ Spec (CommRingCat.of ℂ)) |
| (l : ℕ) (hl : 0 < l) (U : Fin l → X.Opens) |
| (hU : ∀ i, IsAffineOpen (U i)) (hcover : (⨆ i, U i) = ⊤) |
| (presentation : ∀ (M : X.Modules) [M.IsFinitePresentation], |
| Nonempty (CoherentFinitePresentation p M)) |
| (M : X.Modules) [M.IsFinitePresentation] (q : ℕ) : |
| letI := Module.compHom (cohomology M q) (baseScalars p) |
| FiniteDimensional ℂ (cohomology M q) := by |
| let := sheafComplexLinear p |
| let F : {N : X.Modules // N.IsFinitePresentation} → X.Modules := Subtype.val |
| let P (i : {N : X.Modules // N.IsFinitePresentation}) : |
| FiniteCohomologyPresentation (K := ℂ) (structureSheaf X) F i := by |
| letI : i.val.IsFinitePresentation := i.property |
| let S := Classical.choice (presentation i.val) |
| exact |
| { kernel := ⟨S.kernel, S.kernel_coherent⟩ |
| middle := S.middle |
| left := S.left |
| right := S.right |
| comp_zero := S.comp_zero |
| shortExact := S.shortExact |
| finite := fun n => by |
| change @Module.Finite ℂ (cohomology S.middle n) _ _ (complexExtModule p S.middle n) |
| rw [complexExtModule_eq] |
| exact S.middle_finite n } |
| have hbound : ∀ i n, l ≤ n → ∀ x : Ext.{1} (structureSheaf X) (F i) n, x = 0 := by |
| intro i n hn x |
| let : i.val.IsFinitePresentation := i.property |
| let : i.val.IsQuasicoherent := |
| (SheafOfModules.IsFinitePresentation.exists_quasicoherentData i.val).choose.isQuasicoherent |
| exact PiExponent.SerreVanishing.ext_eq_zero_of_affine_cover l hl U hU hcover i.val n hn x |
| have hf := ext_finite_of_presentations (K := ℂ) (structureSheaf X) F l hbound P |
| ⟨M, inferInstance⟩ q |
| change @Module.Finite ℂ (cohomology M q) _ _ (complexExtModule p M q) at hf |
| rw [complexExtModule_eq] at hf |
| exact hf |
|
|
| end |
| end PiExponent.CoherentExtFiniteness |
|
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| end OAI |
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