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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 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 | import OAI.Algebra.AuslanderReiten.Tensor.BalancedTensorResolution
import OAI.Algebra.AuslanderReiten.Tensor.BalancedTensorCanonicalEndpoint
import OAI.Algebra.AuslanderReiten.Complete.CompleteFunctorCokernelAction
import OAI.Algebra.AuslanderReiten.Tensor.BalancedTensorNative
namespace OAI
noncomputable section
open CategoryTheory CategoryTheory.Abelian CategoryTheory.Limits
open CochainComplex.HomComplex
open scoped ModuleCat.Algebra
namespace ArExplicit.BalancedTensor
open CompleteComparison
universe u
variable {k R F : Type u} [Field k] [CharP k 2] [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] [SMulCommClass 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]
variable [HasDerivedCategory (ModuleCat.{u} R)]
attribute [local instance] moduleCatBaseModule moduleCatBaseTower
variable [PreservesFiniteLimits (functor (k := k) (R := R) (S := R) (F := F))]
variable [PreservesFiniteColimits (functor (k := k) (R := R) (S := R) (F := F))]
variable [(functor (k := k) (R := R) (S := R) (F := F)).PreservesProjectiveObjects]
variable (τ : FrobeniusTrace k R) (M : FiniteLeftModule k R)
variable (e : BalancedTensor k R F M.carrier ≃ₗ[R] M.carrier)
def nativeEvaluationIso :
(functor (k := k) (R := R) (S := R) (F := F)).obj M.carrier ≅ M.carrier :=
nativeObjectIso (k := k) (R := R) (S := R) (F := F) M.carrier ≪≫ e.toModuleIso
def completeActionEndpoint : ModuleCat.of R (CochainCokernel (M.completeCochain τ)) ≅
(functor (k := k) (R := R) (S := R) (F := F)).obj
(ModuleCat.of R (CochainCokernel (M.completeCochain τ))) :=
(M.completeCochainCokernelEquiv τ).toModuleIso ≪≫ (nativeEvaluationIso M e).symm ≪≫
(functor (k := k) (R := R) (S := R) (F := F)).mapIso
(M.completeCochainCokernelEquiv τ).symm.toModuleIso
def completeActionCokernelIso :
ModuleCat.of R (CochainCokernel (completeTensor (F := F) τ M)) ≅
ModuleCat.of R (CochainCokernel (M.completeCochain τ)) :=
rawFunctorEndpointIso (functor (k := k) (R := R) (S := R) (F := F)) (M.completeCochain τ)
(M.completeCochain_totallyAcyclic τ).to_chain
(completeTensor_totallyAcyclic (F := F) τ M).to_chain ≪≫
(completeActionEndpoint τ M e).symm
def completeActionIn : M.completeCochain τ ⟶ completeTensor (F := F) τ M :=
cokernelCochainLift (M.completeCochain_totallyAcyclic τ)
(completeTensor_totallyAcyclic (F := F) τ M)
(completeActionCokernelIso τ M e).inv.hom
def completeActionOut : completeTensor (F := F) τ M ⟶ M.completeCochain τ :=
cokernelCochainLift (completeTensor_totallyAcyclic (F := F) τ M)
(M.completeCochain_totallyAcyclic τ) (completeActionCokernelIso τ M e).hom.hom
omit [CharP k 2] [SMulCommClass k Rᵐᵒᵖ F]
[HasDerivedCategory (ModuleCat.{u} R)] in
theorem completeAction_inverse :
(HomotopyCategory.quotient _ _).map
(completeActionIn τ M e ≫ completeActionOut τ M e) = 𝟙 _ := by
exact cokernelCochainLift_homotopy_inverse (M.completeCochain_totallyAcyclic τ)
(completeTensor_totallyAcyclic (F := F) τ M)
(completeActionCokernelIso τ M e).symm.toLinearEquiv
omit [CharP k 2] [SMulCommClass k Rᵐᵒᵖ F]
[HasDerivedCategory (ModuleCat.{u} R)] in
theorem completeAction_inverse' :
(HomotopyCategory.quotient _ _).map
(completeActionOut τ M e ≫ completeActionIn τ M e) = 𝟙 _ := by
exact cokernelCochainLift_homotopy_inverse (completeTensor_totallyAcyclic (F := F) τ M)
(M.completeCochain_totallyAcyclic τ) (completeActionCokernelIso τ M e).toLinearEquiv
omit [SMulCommClass k Rᵐᵒᵖ F] in
theorem completeAction_scalar (n : ℕ) (c : k)
(hs : ∀ z : Ext M.carrier M.carrier (n+1),
selfExtIso (k := k) (nativeEvaluationIso M e) (n+1)
(z.mapExactFunctor (functor (k := k) (R := R) (S := R) (F := F))) = c • z)
(x : CohomologyClass (M.completeCochain τ) (M.completeCochain τ) ((n+1 : ℕ) : ℤ)) :
fullHomPostcomp (completeActionOut τ M e) _
(fullHomPrecomp (completeActionIn τ M e) _
(mapFullHom (functor (k := k) (R := R) (S := R) (F := F)) _ _ _ x)) = c • x := by
apply fullHom_rawFunctor_scalar (functor (k := k) (R := R) (S := R) (F := F))
(M.completeCochain τ) (M.completeCochain_totallyAcyclic τ).to_chain
(completeTensor_totallyAcyclic (F := F) τ M).to_chain
(completeActionIn τ M e) (completeActionOut τ M e) (completeActionEndpoint τ M e)
· exact cokernelCochainLift_map _ _ _
· exact cokernelCochainLift_map _ _ _
· exact exactFunctor_scalar_transport (functor (k := k) (R := R) (S := R) (F := F))
(M.completeCochainCokernelEquiv τ).toModuleIso (nativeEvaluationIso M e).symm (n+1) c hs
omit [CharP k 2] [SMulCommClass k Rᵐᵒᵖ F]
[HasDerivedCategory (ModuleCat.{u} R)] in
theorem completeActionCokernelIso_eq : completeActionCokernelIso τ M e =
(((tensorCokernelEquiv (F := F) τ M).trans e).trans
(M.completeCochainCokernelEquiv τ).symm).toModuleIso := by
rw [completeActionCokernelIso, rawFunctorEndpointIso_eq]
apply Iso.ext
have ht := congrArg (fun f =>
(cochainCokernelEquiv (k := k) (F := F) (S := R) (M.completeCochain τ)).toModuleIso.hom ≫
f ≫ e.toModuleIso.hom ≫ (M.completeCochainCokernelEquiv τ).toModuleIso.inv)
(nativeObjectIso_natural (k := k) (F := F) (S := R)
(M.completeCochainCokernelEquiv τ).toLinearMap)
exact ht
omit [CharP k 2] [SMulCommClass k Rᵐᵒᵖ F]
[HasDerivedCategory (ModuleCat.{u} R)] in
theorem completeActionOut_quotient :
(HomotopyCategory.quotient _ _).map (completeActionOut τ M e) =
(completeTensorIso τ M M e).hom := by
unfold completeActionOut
rw [completeActionCokernelIso_eq]
exact cokernelCochainLift_quotient _ _ _
end ArExplicit.BalancedTensor
end
end OAI
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