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ae00908 | 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 | import OAI.Algebra.Finitistic.Tensor.BalancedFreeLeft
import OAI.Algebra.Finitistic.Ordinary.OrdinaryProjectiveTensor
import OAI.Algebra.Finitistic.Bimodule.BimoduleTensorFunctor
namespace OAI
noncomputable section
open CategoryTheory
namespace LittleFinitistic.BalancedTensor
universe u
variable {k A M M' M'' N N' : Type u} [Field k] [Ring A] [Algebra k A]
variable [AddCommGroup M] [Module k M] [Module Aᵐᵒᵖ M] [IsScalarTower k Aᵐᵒᵖ M]
variable [AddCommGroup M'] [Module k M'] [Module Aᵐᵒᵖ M'] [IsScalarTower k Aᵐᵒᵖ M']
variable [AddCommGroup M''] [Module k M''] [Module Aᵐᵒᵖ M''] [IsScalarTower k Aᵐᵒᵖ M'']
variable [AddCommGroup N] [Module k N] [Module A N] [IsScalarTower k A N]
variable [AddCommGroup N'] [Module k N'] [Module A N'] [IsScalarTower k A N']
omit [IsScalarTower k Aᵐᵒᵖ M] in
theorem mapRightTo_split (i : N →ₗ[A] N') (p : N' →ₗ[A] N)
(hp : p.comp i = LinearMap.id) (z : Tensor k A M N) :
mapRightTo p (mapRightTo i z) = z := by
induction z using BalancedTensor.induction_on with
| hz => simp
| ht m n =>
rw [mapRightTo_mk,mapRightTo_mk]
exact congrArg (mk k A M N m) (LinearMap.congr_fun hp n)
| ha x y hx hy => simp only [map_add,hx,hy]
theorem leftTensor_exact_of_split (i : N →ₗ[A] N') (p : N' →ₗ[A] N)
(hp : p.comp i = LinearMap.id) (f : M →ₗ[Aᵐᵒᵖ] M') (g : M' →ₗ[Aᵐᵒᵖ] M'')
(h : Function.Exact f g)
(h' : Function.Exact (mapLeft (k := k) N' f) (mapLeft (k := k) N' g)) :
Function.Exact (mapLeft (k := k) N f) (mapLeft (k := k) N g) := by
intro z
constructor
· intro hz
have hi : mapLeft N' g (mapRightTo i z) = 0 := by
rw [mapLeft_mapRightTo,hz,map_zero]
obtain ⟨y,hy⟩ := (h' _).mp hi
refine ⟨mapRightTo p y, ?_⟩
rw [mapLeft_mapRightTo,hy,mapRightTo_split i p hp]
· rintro ⟨y,rfl⟩
induction y using BalancedTensor.induction_on with
| hz => simp
| ht m n => rw [mapLeft_mk,mapLeft_mk,(h (f m)).mpr ⟨m,rfl⟩,map_zero,LinearMap.zero_apply]
| ha x y hx hy => simp only [map_add,hx,hy,add_zero]
/-- Projectivity of the left input preserves every exact sequence in the right variable. -/
theorem projectiveLeftTensor_exact [Module.Projective A N]
(f : M →ₗ[Aᵐᵒᵖ] M') (g : M' →ₗ[Aᵐᵒᵖ] M'') (h : Function.Exact f g) :
Function.Exact (mapLeft (k := k) N f) (mapLeft (k := k) N g) := by
obtain ⟨i,hi⟩ := (Module.projective_def' (R := A) (P := N)).mp inferInstance
exact leftTensor_exact_of_split i (Finsupp.linearCombination A id) hi f g h
(FreeLeft.tensor_exact (k := k) N f g h)
end LittleFinitistic.BalancedTensor
namespace LittleFinitistic.BimoduleBalancedTensor
open ExternalTensor.Section4
universe u
variable (k B N : Type u) [Field k] [Ring B] [Algebra k B]
variable [AddCommGroup N] [Module k N] [Module B N] [IsScalarTower k B N]
variable [Module.Projective B N]
/-- Exactness of tensoring genuine bimodules with a projective left module,
with the result still in B-Mod. -/
theorem projectiveInput_mapsExact
(S : ShortComplex (ModuleCat.{u} (Enveloping k B))) (hS : S.Exact) :
(S.map (functor k B N)).Exact := by
rw [ShortComplex.moduleCat_exact_iff]
have h : Function.Exact
(((forgetRight k B).map S.f).hom : Input k B S.X₁ →ₗ[Bᵐᵒᵖ] Input k B S.X₂)
(((forgetRight k B).map S.g).hom : Input k B S.X₂ →ₗ[Bᵐᵒᵖ] Input k B S.X₃) := by
intro x
constructor
· exact (ShortComplex.moduleCat_exact_iff S).mp hS x
· rintro ⟨y,rfl⟩
exact congrArg (fun t => t.hom y) S.zero
intro z hz
exact ((BalancedTensor.projectiveLeftTensor_exact (k := k) (N := N)
(((forgetRight k B).map S.f).hom : Input k B S.X₁ →ₗ[Bᵐᵒᵖ] Input k B S.X₂)
(((forgetRight k B).map S.g).hom : Input k B S.X₂ →ₗ[Bᵐᵒᵖ] Input k B S.X₃) h) z).mp hz
theorem projectiveInput_preservesHomology : (functor k B N).PreservesHomology :=
Functor.preservesHomology_of_map_exact _ (projectiveInput_mapsExact k B N)
/-- An actual bimodule augmentation quasi-isomorphism remains one on each projective input row. -/
theorem projectiveInput_quasiIso {P Q : BiComplex k B} (ε : P ⟶ Q) [QuasiIso ε] :
QuasiIso (((functor k B N).mapHomologicalComplex (.up ℤ)).map ε) := by
have := projectiveInput_preservesHomology k B N
infer_instance
end LittleFinitistic.BimoduleBalancedTensor
end
end OAI
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