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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 | import OAI.Algebra.AuslanderReiten.Tensor.BalancedTensorComplete
import OAI.Algebra.AuslanderReiten.Tensor.BalancedTensorMapEquiv
import OAI.Algebra.AuslanderReiten.Cones.CochainCokernelIso
import OAI.Algebra.AuslanderReiten.Duality.FiniteStableResolutionIso
namespace OAI
noncomputable section
namespace ArExplicit.BalancedTensor
open CategoryTheory 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] [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]
attribute [local instance] moduleCatBaseModule moduleCatBaseTower
variable (τ : FrobeniusTrace k R) (M N : FiniteLeftModule k R)
abbrev completeTensor :=
(((functor (k := k) (R := R) (S := R) (F := F)).mapHomologicalComplex _).obj
(M.completeCochain τ))
def tensorCokernelEquiv : CochainCokernel (completeTensor (F := F) τ M) ≃ₗ[R]
BalancedTensor k R F M.carrier :=
(cochainCokernelEquiv (k := k) (F := F) (S := R) (M.completeCochain τ)).trans
(mapRightEquiv (k := k) (F := F) (S := R) (M.completeCochainCokernelEquiv τ))
omit [SMulCommClass k Rᵐᵒᵖ F] [Module.Finite R F] [Module.Projective R F] in
@[simp] theorem tensorCokernelEquiv_mk (f : F) (x : (M.completeCochain τ).X 0) :
tensorCokernelEquiv (F := F) τ M (Submodule.Quotient.mk (tmul f x)) =
tmul f (M.completeCochainCokernelEquiv τ (Submodule.Quotient.mk x)) := by
rw [tensorCokernelEquiv, LinearEquiv.trans_apply,
cochainCokernelEquiv_mk_tmul, mapRightEquiv_tmul]
omit [SMulCommClass k Rᵐᵒᵖ F] in
theorem completeTensor_totallyAcyclic : TotallyAcyclicCochain (completeTensor (F := F) τ M) := by
let := τ.regular_injective
exact complex_totallyAcyclic (M.completeCochain τ)
(M.completeCochain_totallyAcyclic τ).toExactProjectiveCochain
def completeTensorIso (e : BalancedTensor k R F M.carrier ≃ₗ[R] N.carrier) :
(HomotopyCategory.quotient _ _).obj (completeTensor (F := F) τ M) ≅
(HomotopyCategory.quotient _ _).obj (N.completeCochain τ) :=
rawCokernelHomotopyIso (completeTensor_totallyAcyclic (F := F) τ M)
(N.completeCochain_totallyAcyclic τ)
(((tensorCokernelEquiv (F := F) τ M).trans e).trans (N.completeCochainCokernelEquiv τ).symm)
variable {G : Type u} [AddCommGroup G] [Module k G] [Module Rᵐᵒᵖ G] [Module R G]
variable [IsScalarTower k Rᵐᵒᵖ G] [SMulCommClass 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]
omit [SMulCommClass k Rᵐᵒᵖ F] [Module.Finite R F] [Module.Projective R F] in
omit [SMulCommClass k Rᵐᵒᵖ G] [Module.Finite R G] [Module.Projective R G] in
theorem tensorCokernelEquiv_natural (a : F →ₗ[Rᵐᵒᵖ] G)
(ha : ∀ (r : R) f, a (r • f) = r • a f)
(z : CochainCokernel (completeTensor (F := F) τ M)) :
tensorCokernelEquiv (F := G) τ M
(cochainCokernelMap (complexMap (k := k) a ha (M.completeCochain τ)) z) =
mapLeft (k := k) (M := M.carrier) a (tensorCokernelEquiv (F := F) τ M z) := by
obtain ⟨x,rfl⟩ := Submodule.mkQ_surjective
(LinearMap.range ((completeTensor (F := F) τ M).d (-1) 0).hom) z
induction x using induction_on with
| zero => repeat' erw [map_zero]
| add x y hx hy =>
repeat' erw [map_add]
exact congrArg₂ (· + ·) hx hy
| tmul f m =>
erw [cochainCokernelMap_mk]
change tensorCokernelEquiv (F := G) τ M (Submodule.Quotient.mk (tmul (a f) m)) = _
erw [tensorCokernelEquiv_mk]
variable (N' : FiniteLeftModule k R)
omit [SMulCommClass k Rᵐᵒᵖ F] [SMulCommClass k Rᵐᵒᵖ G] in
theorem completeTensorIso_natural
(a : F →ₗ[Rᵐᵒᵖ] G) (ha : ∀ (r : R) f, a (r • f) = r • a f)
(e : BalancedTensor k R F M.carrier ≃ₗ[R] N.carrier)
(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)) ≫
(completeTensorIso τ M N' e').hom =
(completeTensorIso τ M N e).hom ≫
(HomotopyCategory.quotient _ _).map (FiniteLeftModule.resolutionMap τ b) := by
apply rawCokernelHomotopyIso_natural
rw [FiniteLeftModule.resolutionMap_cokernel]
apply LinearMap.ext
intro z
change (N'.completeCochainCokernelEquiv τ).symm
(e' (tensorCokernelEquiv (F := G) τ M
(cochainCokernelMap (complexMap (k := k) a ha (M.completeCochain τ)) z))) =
(N'.completeCochainCokernelEquiv τ).symm
(b ((N.completeCochainCokernelEquiv τ)
((N.completeCochainCokernelEquiv τ).symm (e (tensorCokernelEquiv (F := F) τ M z)))))
rw [tensorCokernelEquiv_natural, LinearEquiv.apply_symm_apply, hab]
end ArExplicit.BalancedTensor
end
end OAI
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