| import Mathlib |
| import OAI.AlgebraicGeometry.Seshadri.Model |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| |
| section |
|
|
| namespace MaximalSeshadri.Cohomology |
| noncomputable section |
| open CategoryTheory CategoryTheory.Abelian |
|
|
| def alternatingTrunc (a : ℕ → ℤ) (d : ℕ) : ℤ := |
| ∑ i ∈ Finset.range (d+1), (-1 : ℤ)^i * a i |
|
|
| lemma alternatingTrunc_succ (a : ℕ → ℤ) (d : ℕ) : |
| alternatingTrunc a (d+1) = alternatingTrunc a d + (-1 : ℤ)^(d+1) * a (d+1) := by |
| exact Finset.sum_range_succ _ _ |
|
|
| lemma alternating_rank_identity (a b c f g h : ℕ → ℤ) |
| (hB : ∀ n, b n = f n + g n) (hC : ∀ n, c n = g n + h n) |
| (hA0 : a 0 = f 0) (hA : ∀ n, a (n+1) = h n + f (n+1)) (d : ℕ) : |
| alternatingTrunc b d - alternatingTrunc a d - alternatingTrunc c d = |
| (-1 : ℤ)^(d+1) * h d := by |
| induction d with |
| | zero => simp [alternatingTrunc, hB, hC, hA0] |
| | succ d ih => |
| simp only [alternatingTrunc_succ] |
| calc |
| _ = (alternatingTrunc b d - alternatingTrunc a d - alternatingTrunc c d) + |
| (-1 : ℤ)^(d+1) * (b (d+1) - a (d+1) - c (d+1)) := by ring |
| _ = _ := by rw [ih, hB, hC, hA, pow_succ _ (d+1)]; ring |
|
|
| lemma finrank_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] [FiniteDimensional K V] |
| (f : U →ₗ[K] V) (g : V →ₗ[K] W) (hfg : Function.Exact f g) : |
| Module.finrank K V = Module.finrank K f.range + Module.finrank K g.range := by |
| have hh := LinearMap.finrank_range_add_finrank_ker g |
| rw [hfg.linearMap_ker_eq] at hh |
| omega |
|
|
| universe w v u t |
| variable {K : Type t} [Field K] {C : Type u} [Category.{v} C] [Abelian C] |
| [Linear K C] [HasExt.{w} C] |
|
|
| def extEuler (E M : C) (d : ℕ) : ℤ := |
| alternatingTrunc (fun n => Module.finrank K (Ext E M n)) d |
|
|
| variable (E : C) {S : ShortComplex C} (hS : S.ShortExact) |
|
|
| def cohomologyMap₁ (n : ℕ) : Ext E S.X₁ n →ₗ[K] Ext E S.X₂ n := |
| (Ext.mk₀ S.f).postcompOfLinear K E (add_zero n) |
|
|
| def cohomologyMap₂ (n : ℕ) : Ext E S.X₂ n →ₗ[K] Ext E S.X₃ n := |
| (Ext.mk₀ S.g).postcompOfLinear K E (add_zero n) |
|
|
| def cohomologyBoundary (n : ℕ) : Ext E S.X₃ n →ₗ[K] Ext E S.X₁ (n+1) := |
| hS.extClass.postcompOfLinear K E rfl |
|
|
| include hS in |
| lemma cohomology_exact₂ (n : ℕ) : |
| Function.Exact (cohomologyMap₁ (K := K) E (S := S) n) |
| (cohomologyMap₂ (K := K) E (S := S) n) := by |
| intro x |
| constructor |
| · exact Ext.covariant_sequence_exact₂ E hS x |
| · rintro ⟨y, rfl⟩ |
| change (y.comp (Ext.mk₀ S.f) (add_zero n)).comp (Ext.mk₀ S.g) (add_zero n) = 0 |
| simp [Ext.mk₀_comp_mk₀] |
|
|
| lemma cohomology_exact₃ (n : ℕ) : |
| Function.Exact (cohomologyMap₂ (K := K) E (S := S) n) |
| (cohomologyBoundary (K := K) E hS n) := by |
| intro x |
| constructor |
| · exact Ext.covariant_sequence_exact₃ E hS x rfl |
| · rintro ⟨y, rfl⟩ |
| change (y.comp (Ext.mk₀ S.g) (add_zero n)).comp hS.extClass rfl = 0 |
| rw [Ext.comp_assoc_of_second_deg_zero, hS.comp_extClass, Ext.comp_zero] |
|
|
| lemma cohomology_exact₁ (n : ℕ) : |
| Function.Exact (cohomologyBoundary (K := K) E hS n) |
| (cohomologyMap₁ (K := K) E (S := S) (n+1)) := by |
| intro x |
| constructor |
| · exact fun hx => Ext.covariant_sequence_exact₁ E hS x hx rfl |
| · rintro ⟨y, rfl⟩ |
| change (y.comp hS.extClass rfl).comp (Ext.mk₀ S.f) (add_zero (n+1)) = 0 |
| rw [Ext.comp_assoc_of_third_deg_zero, hS.extClass_comp, Ext.comp_zero] |
|
|
| theorem extEuler_add_of_boundary_zero (d : ℕ) |
| (hfinite₁ : ∀ n ≤ d, FiniteDimensional K (Ext E S.X₁ n)) |
| (hfinite₂ : ∀ n ≤ d, FiniteDimensional K (Ext E S.X₂ n)) |
| (hfinite₃ : ∀ n ≤ d, FiniteDimensional K (Ext E S.X₃ n)) |
| (hvan : cohomologyBoundary (K := K) E hS d = 0) : |
| extEuler (K := K) E S.X₂ d = |
| extEuler (K := K) E S.X₁ d + extEuler (K := K) E S.X₃ d := by |
| classical |
| let a (n : ℕ) : ℤ := Module.finrank K (Ext E S.X₁ n) |
| let b (n : ℕ) : ℤ := Module.finrank K (Ext E S.X₂ n) |
| let c (n : ℕ) : ℤ := Module.finrank K (Ext E S.X₃ n) |
| let f (n : ℕ) : ℤ := Module.finrank K (cohomologyMap₁ (K := K) E (S := S) n).range |
| let g (n : ℕ) : ℤ := Module.finrank K (cohomologyMap₂ (K := K) E (S := S) n).range |
| let h (n : ℕ) : ℤ := Module.finrank K (cohomologyBoundary (K := K) E hS n).range |
| have hB (n : ℕ) (hn : n ≤ d) : b n = f n + g n := by |
| let := hfinite₂ n hn |
| dsimp only [b, f, g] |
| exact_mod_cast finrank_middle_of_exact _ _ (cohomology_exact₂ (K := K) E hS n) |
| have hC (n : ℕ) (hn : n ≤ d) : c n = g n + h n := by |
| let := hfinite₃ n hn |
| dsimp only [c, g, h] |
| exact_mod_cast finrank_middle_of_exact _ _ (cohomology_exact₃ (K := K) E hS n) |
| have hA0 : a 0 = f 0 := by |
| have : Mono S.f := hS.mono_f |
| dsimp only [a, f] |
| exact_mod_cast (LinearMap.finrank_range_of_inj |
| (f := cohomologyMap₁ (K := K) E (S := S) 0) |
| (Ext.postcomp_mk₀_injective_of_mono E S.f)).symm |
| have hA (n : ℕ) (hn : n+1 ≤ d) : a (n+1) = h n + f (n+1) := by |
| let := hfinite₁ (n+1) hn |
| dsimp only [a, h, f] |
| exact_mod_cast finrank_middle_of_exact _ _ (cohomology_exact₁ (K := K) E hS n) |
| have hind (n : ℕ) (hn : n ≤ d) : |
| alternatingTrunc b n - alternatingTrunc a n - alternatingTrunc c n = |
| (-1 : ℤ)^(n+1) * h n := by |
| induction n with |
| | zero => simp [alternatingTrunc, hB 0 hn, hC 0 hn, hA0] |
| | succ n ih => |
| have hi := ih (by omega) |
| simp only [alternatingTrunc_succ] |
| calc |
| _ = (alternatingTrunc b n - alternatingTrunc a n - alternatingTrunc c n) + |
| (-1 : ℤ)^(n+1) * (b (n+1) - a (n+1) - c (n+1)) := by ring |
| _ = _ := by rw [hi, hB (n+1) hn, hC (n+1) hn, hA n hn, pow_succ _ (n+1)]; ring |
| have hd : h d = 0 := by |
| dsimp only [h] |
| rw [hvan, LinearMap.range_zero] |
| rw [finrank_bot] |
| rfl |
| have hh := hind d le_rfl |
| rw [hd, mul_zero] at hh |
| change alternatingTrunc b d = alternatingTrunc a d + alternatingTrunc c d |
| omega |
|
|
| include hS in |
|
|
| theorem extEuler_add (d : ℕ) |
| (hfinite₁ : ∀ n ≤ d, FiniteDimensional K (Ext E S.X₁ n)) |
| (hfinite₂ : ∀ n ≤ d, FiniteDimensional K (Ext E S.X₂ n)) |
| (hfinite₃ : ∀ n ≤ d, FiniteDimensional K (Ext E S.X₃ n)) |
| [Subsingleton (Ext E S.X₁ (d+1))] : |
| extEuler (K := K) E S.X₂ d = |
| extEuler (K := K) E S.X₁ d + extEuler (K := K) E S.X₃ d := |
| extEuler_add_of_boundary_zero E hS d hfinite₁ hfinite₂ hfinite₃ (Subsingleton.elim _ _) |
|
|
| end |
| end MaximalSeshadri.Cohomology |
|
|
| namespace MaximalSeshadri.Geometry |
| noncomputable section |
| open CategoryTheory CategoryTheory.Abelian AlgebraicGeometry |
| open scoped AlgebraicGeometry |
| variable {X : Scheme.{0}} (f : X ⟶ Spec (CommRingCat.of ℂ)) |
|
|
| @[instance_reducible] |
| def sheafComplexLinear : Linear ℂ X.Modules where |
| homModule M N := Module.compHom _ (baseScalars f) |
| smul_comp M N P r g h := by |
| change ((baseScalars f r) • g) ≫ h = (baseScalars f r) • (g ≫ h) |
| exact Linear.smul_comp _ _ _ _ _ _ |
| comp_smul M N P g r h := by |
| change g ≫ ((baseScalars f r) • h) = (baseScalars f r) • (g ≫ h) |
| exact Linear.comp_smul _ _ _ _ _ _ |
|
|
| @[instance_reducible] |
| def complexExtModule (M : X.Modules) (n : ℕ) : Module ℂ (cohomology M n) := |
| letI := sheafComplexLinear f |
| inferInstance |
|
|
| lemma complexExtModule_eq (M : X.Modules) (n : ℕ) : |
| complexExtModule f M n = Module.compHom (cohomology M n) (baseScalars f) := by |
| apply Module.ext' |
| intro r x |
| let := sheafComplexLinear f |
| change r • x = (baseScalars f r) • x |
| rw [Ext.smul_eq_comp_mk₀ (R := ℂ), Ext.smul_eq_comp_mk₀ (R := Γ(X, ⊤))] |
| rfl |
|
|
| lemma eulerCharacteristic_eq_extEuler (M : X.Modules) (d : ℕ) : |
| eulerCharacteristic f d M = |
| letI := sheafComplexLinear f |
| Cohomology.extEuler (K := ℂ) (structureSheaf X) M d := by |
| unfold eulerCharacteristic Cohomology.extEuler Cohomology.alternatingTrunc |
| apply Finset.sum_congr rfl |
| intro n hn |
| have hh := complexExtModule_eq f M n |
| unfold cohomologyDimension |
| rw [← hh] |
|
|
| theorem eulerCharacteristic_add {S : ShortComplex X.Modules} (hS : S.ShortExact) |
| (d : ℕ) |
| (hfinite₁ : ∀ n ≤ d, letI := Module.compHom (cohomology S.X₁ n) (baseScalars f) |
| FiniteDimensional ℂ (cohomology S.X₁ n)) |
| (hfinite₂ : ∀ n ≤ d, letI := Module.compHom (cohomology S.X₂ n) (baseScalars f) |
| FiniteDimensional ℂ (cohomology S.X₂ n)) |
| (hfinite₃ : ∀ n ≤ d, letI := Module.compHom (cohomology S.X₃ n) (baseScalars f) |
| FiniteDimensional ℂ (cohomology S.X₃ n)) |
| [Subsingleton (cohomology S.X₁ (d+1))] : |
| eulerCharacteristic f d S.X₂ = |
| eulerCharacteristic f d S.X₁ + eulerCharacteristic f d S.X₃ := by |
| let := sheafComplexLinear f |
| rw [eulerCharacteristic_eq_extEuler, eulerCharacteristic_eq_extEuler, |
| eulerCharacteristic_eq_extEuler] |
| apply Cohomology.extEuler_add (K := ℂ) (structureSheaf X) hS d |
| · intro n hn |
| have hh := complexExtModule_eq f S.X₁ n |
| change @FiniteDimensional ℂ (cohomology S.X₁ n) _ _ (complexExtModule f S.X₁ n) |
| rw [hh] |
| exact hfinite₁ n hn |
| · intro n hn |
| have hh := complexExtModule_eq f S.X₂ n |
| change @FiniteDimensional ℂ (cohomology S.X₂ n) _ _ (complexExtModule f S.X₂ n) |
| rw [hh] |
| exact hfinite₂ n hn |
| · intro n hn |
| have hh := complexExtModule_eq f S.X₃ n |
| change @FiniteDimensional ℂ (cohomology S.X₃ n) _ _ (complexExtModule f S.X₃ n) |
| rw [hh] |
| exact hfinite₃ n hn |
|
|
| end |
| end MaximalSeshadri.Geometry |
| end |
|
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|
|
| end |
| end |
|
|
| end OAI |
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|
