| import OAI.Algebra.AuslanderReiten.Modules.FiniteModuleNaturality |
| import OAI.Algebra.AuslanderReiten.HomComplex.FullHomLinear |
|
|
| namespace OAI |
|
|
| open CategoryTheory |
| open scoped ModuleCat.Algebra |
|
|
| namespace ArExplicit |
| universe u |
| variable {k R : Type u} [Field k] [Ring R] [Algebra k R] |
|
|
| namespace CompleteComparison |
| variable {K L : CochainComplex (ModuleCat.{u} R) ℤ} |
|
|
| theorem cochainCokernelMap_smul (a : k) (f : K ⟶ L) : |
| cochainCokernelMap (a • f) = a • cochainCokernelMap f := by |
| apply LinearMap.ext |
| intro z |
| obtain ⟨x, rfl⟩ := (LinearMap.range (K.d (-1) 0).hom).mkQ_surjective z |
| change (LinearMap.range (L.d (-1) 0).hom).mkQ (a • (f.f 0).hom x) = |
| a • (LinearMap.range (L.d (-1) 0).hom).mkQ ((f.f 0).hom x) |
| exact ((LinearMap.range (L.d (-1) 0).hom).mkQ.restrictScalars k).map_smul a _ |
| end CompleteComparison |
|
|
| namespace FiniteLeftModule |
| open CompleteComparison |
| variable [Module.Finite k R] (τ : FrobeniusTrace k R) |
| variable {M N : FiniteLeftModule k R} |
|
|
| theorem resolutionCokernelMap_smul (a : k) (f : M.carrier →ₗ[R] N.carrier) : |
| resolutionCokernelMap τ (a • f) = a • resolutionCokernelMap τ f := by |
| apply LinearMap.ext |
| intro x |
| exact ((N.completeCochainCokernelEquiv τ).symm.toLinearMap.restrictScalars k).map_smul a _ |
|
|
| theorem resolutionMap_smul (a : k) (f : M.carrier →ₗ[R] N.carrier) : |
| (HomotopyCategory.quotient _ _).map (resolutionMap τ (a • f)) = |
| a • (HomotopyCategory.quotient _ _).map (resolutionMap τ f) := by |
| rw [← Functor.map_smul] |
| apply (cochainHomotopyRawEquiv (M.completeCochain_totallyAcyclic τ).to_chain |
| (N.completeCochain_totallyAcyclic τ).to_chain.toExactProjectiveComplex).injective |
| rw [cochainHomotopyRawEquiv_map, cochainHomotopyRawEquiv_map, |
| resolutionMap_cokernel, cochainCokernelMap_smul, resolutionMap_cokernel, |
| resolutionCokernelMap_smul] |
| end FiniteLeftModule |
| end ArExplicit |
|
|
| end OAI |
|
|
