File size: 3,849 Bytes
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 | import OAI.Algebra.AuslanderReiten.Tensor.BalancedTensorCategory
import OAI.Algebra.AuslanderReiten.Cones.CochainCokernel
namespace OAI
noncomputable section
namespace ArExplicit.BalancedTensor
open CategoryTheory CompleteComparison
universe u
variable {k R S F : Type u} [Field k] [Ring R] [Ring S] [Algebra k R]
variable [AddCommGroup F] [Module k F] [Module Rᵐᵒᵖ F] [Module S F]
variable [IsScalarTower k Rᵐᵒᵖ F]
variable [SMulCommClass k S F] [SMulCommClass S Rᵐᵒᵖ F]
variable [Module.Finite Rᵐᵒᵖ F] [Module.Projective Rᵐᵒᵖ F]
variable {L M : Type u} [AddCommGroup L] [AddCommGroup M]
variable [Module k L] [Module k M] [Module R L] [Module R M]
variable [IsScalarTower k R L] [IsScalarTower k R M]
variable [SMulCommClass k R M]
def cokernelEquiv (d : L →ₗ[R] M) :
(BalancedTensor k R F M ⧸ LinearMap.range (mapRight (k := k) (F := F) (S := S) d)) ≃ₗ[S]
BalancedTensor k R F (M ⧸ LinearMap.range d) :=
(mapRight_exact (k := k) (F := F) (S := S) d (LinearMap.range d).mkQ
(LinearMap.exact_map_mkQ_range d)).linearEquivOfSurjective
(mapRight_surjective (k := k) (F := F) (S := S) _ (Submodule.mkQ_surjective _))
theorem cokernelEquiv_mk (d : L →ₗ[R] M) (x : BalancedTensor k R F M) :
cokernelEquiv (k := k) (F := F) (S := S) d (Submodule.Quotient.mk x) =
mapRight (k := k) (F := F) (S := S) (LinearMap.range d).mkQ x := by
apply (cokernelEquiv (k := k) (F := F) (S := S) d).symm.injective
rw [LinearEquiv.symm_apply_apply]
exact (Function.Exact.linearEquivOfSurjective_symm_apply _ _ x).symm
@[simp] theorem cokernelEquiv_mk_tmul (d : L →ₗ[R] M) (f : F) (m : M) :
cokernelEquiv (k := k) (F := F) (S := S) d (Submodule.Quotient.mk (tmul f m)) =
tmul f (Submodule.Quotient.mk m) := by
rw [cokernelEquiv_mk, mapRight_tmul]
rfl
attribute [local instance] moduleCatBaseModule moduleCatBaseTower
def cochainCokernelEquiv (P : CochainComplex (ModuleCat.{u} R) ℤ) :
CochainCokernel (((functor (k := k) (R := R) (S := S) (F := F)).mapHomologicalComplex _).obj P) ≃ₗ[S]
BalancedTensor k R F (CochainCokernel P) :=
cokernelEquiv (k := k) (F := F) (S := S) (P.d (-1) 0).hom
@[simp] theorem cochainCokernelEquiv_mk_tmul
(P : CochainComplex (ModuleCat.{u} R) ℤ) (f : F) (x : P.X 0) :
cochainCokernelEquiv (k := k) (F := F) (S := S) P
(Submodule.Quotient.mk (tmul f x)) = tmul f (Submodule.Quotient.mk x) :=
cokernelEquiv_mk_tmul (k := k) (F := F) (S := S) (P.d (-1) 0).hom f x
variable {G : Type u} [AddCommGroup G] [Module k G] [Module Rᵐᵒᵖ G] [Module S G]
variable [IsScalarTower k Rᵐᵒᵖ G]
variable [SMulCommClass k S G] [SMulCommClass S Rᵐᵒᵖ G]
variable [Module.Finite Rᵐᵒᵖ G] [Module.Projective Rᵐᵒᵖ G]
theorem cochainCokernelEquiv_natural
(a : F →ₗ[Rᵐᵒᵖ] G) (ha : ∀ (s : S) f, a (s • f) = s • a f)
(P : CochainComplex (ModuleCat.{u} R) ℤ)
(z : CochainCokernel (((functor (k := k) (R := R) (S := S) (F := F)).mapHomologicalComplex _).obj P)) :
cochainCokernelEquiv (k := k) (F := G) (S := S) P
(cochainCokernelMap (complexMap (k := k) a ha P) z) =
mapLeft (k := k) (M := CochainCokernel P) a
(cochainCokernelEquiv (k := k) (F := F) (S := S) P z) := by
obtain ⟨x,rfl⟩ := Submodule.mkQ_surjective
(LinearMap.range (((((functor (k := k) (R := R) (S := S) (F := F)).mapHomologicalComplex _).obj P).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 cochainCokernelEquiv (k := k) (F := G) (S := S) P
(Submodule.Quotient.mk (tmul (a f) m)) = _
erw [cochainCokernelEquiv_mk_tmul]
end ArExplicit.BalancedTensor
end
end OAI
|