openAI-math-dataset / lean /OAI /Algebra /AuslanderReiten /Resolutions /FiniteResolutionAdditive.lean
| import OAI.Algebra.AuslanderReiten.Resolutions.FiniteResolutionLinear | |
| 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_add (f g : K ⟶ L) : | |
| cochainCokernelMap (f+g) = cochainCokernelMap f + cochainCokernelMap g := by | |
| apply LinearMap.ext | |
| intro z | |
| obtain ⟨x, rfl⟩ := (LinearMap.range (K.d (-1) 0).hom).mkQ_surjective z | |
| exact (LinearMap.range (L.d (-1) 0).hom).mkQ.map_add _ _ | |
| end CompleteComparison | |
| namespace FiniteLeftModule | |
| open CompleteComparison | |
| variable [Module.Finite k R] (τ : FrobeniusTrace k R) | |
| variable {M N : FiniteLeftModule k R} | |
| theorem resolutionCokernelMap_add (f g : M.carrier →ₗ[R] N.carrier) : | |
| resolutionCokernelMap τ (f+g) = resolutionCokernelMap τ f + resolutionCokernelMap τ g := by | |
| ext x | |
| exact (N.completeCochainCokernelEquiv τ).symm.map_add _ _ | |
| theorem resolutionMap_add (f g : M.carrier →ₗ[R] N.carrier) : | |
| (HomotopyCategory.quotient _ _).map (resolutionMap τ (f+g)) = | |
| (HomotopyCategory.quotient _ _).map (resolutionMap τ f) + | |
| (HomotopyCategory.quotient _ _).map (resolutionMap τ g) := by | |
| rw [← Functor.map_add] | |
| apply (cochainHomotopyRawEquiv (M.completeCochain_totallyAcyclic τ).to_chain | |
| (N.completeCochain_totallyAcyclic τ).to_chain.toExactProjectiveComplex).injective | |
| rw [cochainHomotopyRawEquiv_map, cochainHomotopyRawEquiv_map, | |
| resolutionMap_cokernel, cochainCokernelMap_add, resolutionMap_cokernel, | |
| resolutionMap_cokernel, resolutionCokernelMap_add] | |
| end FiniteLeftModule | |
| end ArExplicit | |
| end OAI | |