| import OAI.Algebra.AuslanderReiten.HomComplex.FiniteFullHomExt | |
| import OAI.Algebra.AuslanderReiten.HomComplex.TensorFullHomZero | |
| import OAI.Algebra.AuslanderReiten.Ext.NativeExtZero | |
| namespace OAI | |
| noncomputable section | |
| open CategoryTheory CategoryTheory.Abelian | |
| open CochainComplex CochainComplex.HomComplex | |
| open scoped ModuleCat.Algebra | |
| namespace ArExplicit.A | |
| universe u | |
| variable {k : Type u} [Field k] [CharP k 2] (q : k) | |
| open CompleteComparison | |
| variable [HasDerivedCategory (ModuleCat.{u} (A k q))] | |
| def simpleExtZeroEquiv (hq1 : 1+q ≠ 0) : k ≃ₗ[k] Ext (Simple q) (Simple q) 0 := | |
| (simpleEndEquiv q hq1).symm.trans (nativeExtZeroEquiv (Simple q) (Simple q)) | |
| omit [CharP k 2] [HasDerivedCategory (ModuleCat.{u} (A k q))] in | |
| @[simp] | |
| theorem simpleExtZeroEquiv_one (hq1 : 1+q ≠ 0) : | |
| simpleExtZeroEquiv q hq1 1 = Ext.mk₀ (𝟙 (Simple q)) := by | |
| unfold simpleExtZeroEquiv | |
| rw [LinearEquiv.trans_apply, nativeExtZeroEquiv_apply] | |
| congr 1 | |
| apply ModuleCat.hom_ext | |
| change (1:k) • (LinearMap.id : Simple q →ₗ[A k q] Simple q) = LinearMap.id | |
| exact one_smul k _ | |
| omit [HasDerivedCategory (ModuleCat.{u} (A k q))] in | |
| theorem fullHomZero_eq_scalar (hq1 : 1+q ≠ 0) | |
| (x : CohomologyClass (simpleComplete q) (simpleComplete q) 0) : | |
| x = fullHomZeroScalarEquiv q hq1 x • fullHomIdentity (simpleComplete q) := by | |
| apply (fullHomZeroScalarEquiv q hq1).injective | |
| rw [map_smul] | |
| change _ = _ • fullHomZeroScalarEquiv q hq1 (CohomologyClass.mk (Cocycle.ofHom (𝟙 _))) | |
| rw [fullHomZeroScalarEquiv_identity] | |
| exact (mul_one _).symm | |
| def fullNonnegativeExtEquiv (hq1 : 1+q ≠ 0) : (n : ℕ) → | |
| CohomologyClass (simpleComplete q) (simpleComplete q) (n:ℤ) ≃ₗ[k] Ext (Simple q) (Simple q) n | |
| | 0 => (fullHomZeroScalarEquiv q hq1).trans (simpleExtZeroEquiv q hq1) | |
| | n+1 => FiniteLeftModule.fullPositiveModuleExtEquiv (AFrobeniusTrace k q) (simpleFiniteModule q) n | |
| omit [HasDerivedCategory (ModuleCat.{u} (A k q))] in | |
| @[simp] | |
| theorem fullNonnegativeExtEquiv_identity (hq1 : 1+q ≠ 0) : | |
| fullNonnegativeExtEquiv q hq1 0 (fullHomIdentity (simpleComplete q)) = Ext.mk₀ (𝟙 (Simple q)) := by | |
| change simpleExtZeroEquiv q hq1 (fullHomZeroScalarEquiv q hq1 | |
| (CohomologyClass.mk (Cocycle.ofHom (𝟙 _)))) = _ | |
| rw [fullHomZeroScalarEquiv_identity, simpleExtZeroEquiv_one] | |
| theorem fullNonnegativeExtEquiv_product_of_eq (hq1 : 1+q ≠ 0) (n m r : ℕ) (hr : n+m=r) | |
| (x : CohomologyClass (simpleComplete q) (simpleComplete q) (n:ℤ)) | |
| (y : CohomologyClass (simpleComplete q) (simpleComplete q) (m:ℤ)) : | |
| fullNonnegativeExtEquiv q hq1 r | |
| (TateKoszul.degreeEquiv (k := k) (CohomologyClass (simpleComplete q) (simpleComplete q)) | |
| (show (n:ℤ)+(m:ℤ)=(r:ℤ) by omega) (fullHomProduct x y)) = | |
| (fullNonnegativeExtEquiv q hq1 n x).comp (fullNonnegativeExtEquiv q hq1 m y) hr := by | |
| cases n with | |
| | zero => | |
| obtain rfl : r=m := by omega | |
| rw [fullHomZero_eq_scalar q hq1 x] | |
| simp only [fullHomProduct_smul_left, map_smul, | |
| fullNonnegativeExtEquiv_identity, Ext.smul_comp, Ext.mk₀_id_comp] | |
| erw [fullHomProduct_identity_left] | |
| | succ n => | |
| cases m with | |
| | zero => | |
| obtain rfl : r=n+1 := by omega | |
| rw [fullHomZero_eq_scalar q hq1 y] | |
| simp only [fullHomProduct_smul_right, map_smul, | |
| fullNonnegativeExtEquiv_identity, Ext.comp_smul, Ext.comp_mk₀_id] | |
| erw [fullHomProduct_identity_right] | |
| | succ m => | |
| obtain rfl : r=n+1+m+1 := by omega | |
| exact FiniteLeftModule.fullPositiveModuleExtEquiv_product (AFrobeniusTrace k q) | |
| (simpleFiniteModule q) n m x y | |
| theorem fullNonnegativeExtEquiv_product (hq1 : 1+q ≠ 0) (n m : ℕ) | |
| (x : CohomologyClass (simpleComplete q) (simpleComplete q) (n:ℤ)) | |
| (y : CohomologyClass (simpleComplete q) (simpleComplete q) (m:ℤ)) : | |
| fullNonnegativeExtEquiv q hq1 (n+m) | |
| (TateKoszul.degreeEquiv (k := k) (CohomologyClass (simpleComplete q) (simpleComplete q)) | |
| (show (n:ℤ)+(m:ℤ)=((n+m:ℕ):ℤ) by omega) (fullHomProduct x y)) = | |
| (fullNonnegativeExtEquiv q hq1 n x).comp (fullNonnegativeExtEquiv q hq1 m y) rfl := | |
| fullNonnegativeExtEquiv_product_of_eq q hq1 n m (n+m) rfl x y | |
| end ArExplicit.A | |
| end | |
| end OAI | |