| import OAI.Algebra.AuslanderReiten.HomComplex.FiniteFullHomZero |
| import OAI.Algebra.AuslanderReiten.HomComplex.PositiveFullHomProducts |
| import OAI.Algebra.AuslanderReiten.HomComplex.FullHomDegreeProducts |
| import OAI.Algebra.AuslanderReiten.Ext.ExtIsoCongr |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open CategoryTheory CategoryTheory.Abelian |
| open CochainComplex CochainComplex.HomComplex |
| open scoped ModuleCat.Algebra |
| namespace ArExplicit.FiniteLeftModule |
| universe u |
| variable {k R : Type u} [Field k] [Ring R] [Algebra k R] [Module.Finite k R] [CharP k 2] |
| variable [HasDerivedCategory (ModuleCat.{u} R)] |
| open CompleteComparison |
| variable (τ : FrobeniusTrace k R) (M : FiniteLeftModule k R) |
|
|
| def fullPositiveModuleExtEquiv (n : ℕ) : |
| CohomologyClass (M.completeCochain τ) (M.completeCochain τ) ((n+1:ℕ):ℤ) ≃ₗ[k] |
| Ext M.carrier M.carrier (n+1) := |
| (fullPositiveExtEquiv (k := k) _ _ (M.completeCochain_totallyAcyclic τ).to_chain |
| (M.completeCochain_totallyAcyclic τ).to_chain.toExactProjectiveComplex n).trans |
| (selfExtIso (k := k) (M.fullCokernelEquiv τ).toModuleIso (n+1)) |
|
|
| theorem fullPositiveModuleExtEquiv_product (n m : ℕ) |
| (x : CohomologyClass (M.completeCochain τ) (M.completeCochain τ) ((n+1:ℕ):ℤ)) |
| (y : CohomologyClass (M.completeCochain τ) (M.completeCochain τ) ((m+1:ℕ):ℤ)) : |
| fullPositiveModuleExtEquiv τ M (n+1+m) |
| (TateKoszul.degreeEquiv (k := k) (CohomologyClass (M.completeCochain τ) (M.completeCochain τ)) |
| (show ((n+1:ℕ):ℤ)+((m+1:ℕ):ℤ)=((n+1+m+1:ℕ):ℤ) by omega) (fullHomProduct x y)) = |
| (fullPositiveModuleExtEquiv τ M n x).comp (fullPositiveModuleExtEquiv τ M m y) (by omega) := by |
| obtain ⟨x,rfl⟩ := CohomologyClass.mk_surjective x |
| obtain ⟨y,rfl⟩ := CohomologyClass.mk_surjective y |
| rw [fullHomProduct_degree_mk _ x y (ResolutionProducts.natCocycleProduct x y) rfl] |
| change selfExtIso _ _ (fullPositiveExtEquiv _ _ _ _ _ _) = |
| (selfExtIso _ _ (fullPositiveExtEquiv _ _ _ _ _ _)).comp |
| (selfExtIso _ _ (fullPositiveExtEquiv _ _ _ _ _ _)) _ |
| rw [← selfExtIso_comp, fullPositiveExtEquiv_product] |
|
|
| end ArExplicit.FiniteLeftModule |
|
|
| end |
|
|
| end OAI |
|
|
