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2c7d975 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 | 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
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