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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 43 44 45 46 47 48 49 50 | 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
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