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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 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 | import OAI.Algebra.AuslanderReiten.Tensor.BalancedTensorResolution
import OAI.Algebra.AuslanderReiten.Complete.CompleteCokernelLifts
namespace OAI
noncomputable section
open CategoryTheory CochainComplex.HomComplex
open scoped ModuleCat.Algebra
namespace ArExplicit.BalancedTensor
open CompleteComparison
universe u
variable {k R F : Type u} [Field k] [Ring R] [Algebra k R] [Module.Finite k R]
variable [AddCommGroup F] [Module k F] [Module Rᵐᵒᵖ F] [Module R F]
variable [IsScalarTower k Rᵐᵒᵖ F]
variable [SMulCommClass k R F] [SMulCommClass R Rᵐᵒᵖ F]
variable [Module.Finite Rᵐᵒᵖ F] [Module.Projective Rᵐᵒᵖ F]
variable [Module.Finite R F] [Module.Projective R F]
attribute [local instance] moduleCatBaseModule moduleCatBaseTower
variable (τ : FrobeniusTrace k R) (M N : FiniteLeftModule k R)
variable (e : BalancedTensor k R F M.carrier ≃ₗ[R] N.carrier)
def comparisonOut : completeTensor (F := F) τ M ⟶ N.completeCochain τ :=
cokernelCochainLift (completeTensor_totallyAcyclic (F := F) τ M)
(N.completeCochain_totallyAcyclic τ)
(((tensorCokernelEquiv (F := F) τ M).trans e).trans
(N.completeCochainCokernelEquiv τ).symm).toLinearMap
def comparisonIn : N.completeCochain τ ⟶ completeTensor (F := F) τ M :=
cokernelCochainLift (N.completeCochain_totallyAcyclic τ)
(completeTensor_totallyAcyclic (F := F) τ M)
(((tensorCokernelEquiv (F := F) τ M).trans e).trans
(N.completeCochainCokernelEquiv τ).symm).symm.toLinearMap
theorem comparisonOut_quotient :
(HomotopyCategory.quotient _ _).map (comparisonOut τ M N e) =
(completeTensorIso τ M N e).hom :=
cokernelCochainLift_quotient _ _ _
theorem comparisonIn_quotient :
(HomotopyCategory.quotient _ _).map (comparisonIn τ M N e) =
(completeTensorIso τ M N e).inv :=
cokernelCochainLift_quotient_inv _ _ _
theorem comparisonOut_bijective (K : CochainComplex (ModuleCat.{u} R) ℤ) (n : ℤ) :
Function.Bijective (fullHomPostcomp (K := K) (comparisonOut τ M N e) n) := by
apply fullHomPostcomp_bijective k n (comparisonOut τ M N e) (comparisonIn τ M N e)
· rw [CategoryTheory.Functor.map_comp, comparisonOut_quotient, comparisonIn_quotient,
Iso.hom_inv_id]
· rw [CategoryTheory.Functor.map_comp, comparisonOut_quotient, comparisonIn_quotient,
Iso.inv_hom_id]
variable {G : Type u} [AddCommGroup G] [Module k G] [Module Rᵐᵒᵖ G] [Module R G]
variable [IsScalarTower k Rᵐᵒᵖ G]
variable [SMulCommClass k R G] [SMulCommClass R Rᵐᵒᵖ G]
variable [Module.Finite Rᵐᵒᵖ G] [Module.Projective Rᵐᵒᵖ G]
variable [Module.Finite R G] [Module.Projective R G]
variable (N' : FiniteLeftModule k R)
theorem comparisonOut_natural
(a : F →ₗ[Rᵐᵒᵖ] G) (ha : ∀ (r : R) f, a (r • f) = r • a f)
(e' : BalancedTensor k R G M.carrier ≃ₗ[R] N'.carrier)
(b : N.carrier →ₗ[R] N'.carrier)
(hab : ∀ x, e' (mapLeft (k := k) (M := M.carrier) a x) = b (e x)) :
(HomotopyCategory.quotient _ _).map
(complexMap (k := k) a ha (M.completeCochain τ) ≫ comparisonOut τ M N' e') =
(HomotopyCategory.quotient _ _).map
(comparisonOut τ M N e ≫ FiniteLeftModule.resolutionMap τ b) := by
rw [CategoryTheory.Functor.map_comp, CategoryTheory.Functor.map_comp,
comparisonOut_quotient, comparisonOut_quotient]
exact completeTensorIso_natural τ M N N' a ha e e' b hab
theorem comparisonOut_fullHom_natural
(a : F →ₗ[Rᵐᵒᵖ] G) (ha : ∀ (r : R) f, a (r • f) = r • a f)
(e' : BalancedTensor k R G M.carrier ≃ₗ[R] N'.carrier)
(b : N.carrier →ₗ[R] N'.carrier)
(hab : ∀ x, e' (mapLeft (k := k) (M := M.carrier) a x) = b (e x))
(K : CochainComplex (ModuleCat.{u} R) ℤ) (n : ℤ)
(x : CohomologyClass K (completeTensor (F := F) τ M) n) :
fullHomPostcomp (comparisonOut τ M N' e') n
(fullHomPostcomp (complexMap (k := k) a ha (M.completeCochain τ)) n x) =
fullHomPostcomp (FiniteLeftModule.resolutionMap τ b) n
(fullHomPostcomp (comparisonOut τ M N e) n x) := by
rw [fullHomPostcomp_comp, fullHomPostcomp_comp]
exact fullHomPostcomp_congr k n (comparisonOut_natural τ M N e N' a ha e' b hab) x
end ArExplicit.BalancedTensor
end
end OAI
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