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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 | import OAI.Algebra.AuslanderReiten.Triangular.TriangularCochain
import OAI.Algebra.AuslanderReiten.Tensor.BalancedTensorCategory
import OAI.Algebra.AuslanderReiten.HomComplex.FullHomTransport
namespace OAI
noncomputable section
namespace ArExplicit.TriangularInduction
open CategoryTheory CochainComplex.HomComplex
universe u
variable {k R F : Type u} [Field k] [Ring R] [Algebra k R]
variable [AddCommGroup F] [Module k F] [Module R F] [Module Rᵐᵒᵖ F]
variable [SMulCommClass k R F] [SMulCommClass R Rᵐᵒᵖ F]
variable [IsScalarTower k R F] [IsScalarTower k Rᵐᵒᵖ F] [SMulCommClass k Rᵐᵒᵖ F]
attribute [local instance] moduleCatBaseModule moduleCatBaseTower
def crossHomAdd (M N : ModuleCat.{u} R) :
(M ⟶ (BalancedTensor.functor (k := k) (R := R) (S := R) (F := F)).obj N) →+
((jFunctor k R F).obj M ⟶ (eFunctor k R F).obj N) where
toFun f := ModuleCat.ofHom (jHom f.hom)
map_zero' := by
apply ModuleCat.hom_ext
apply jHomEquiv.injective
exact map_zero jHomEquiv |>.symm
map_add' f g := by
apply ModuleCat.hom_ext
apply jHomEquiv.injective
ext z
rfl
theorem crossHomAdd_source {M N L : ModuleCat.{u} R} (f : M ⟶ N)
(g : N ⟶ (BalancedTensor.functor (k := k) (R := R) (S := R) (F := F)).obj L) :
crossHomAdd M L (f ≫ g) = (jFunctor k R F).map f ≫ crossHomAdd N L g := by
apply ModuleCat.hom_ext
apply jHomEquiv.injective
rfl
theorem crossHomAdd_target {M N L : ModuleCat.{u} R}
(f : M ⟶ (BalancedTensor.functor (k := k) (R := R) (S := R) (F := F)).obj N)
(g : N ⟶ L) :
crossHomAdd M L (f ≫ (BalancedTensor.functor (k := k) (R := R) (S := R) (F := F)).map g) =
crossHomAdd M N f ≫ (eFunctor k R F).map g := by
apply ModuleCat.hom_ext
apply jHomEquiv.injective
rfl
abbrev tensorCochain (P : CochainComplex (ModuleCat.{u} R) ℤ) :=
((BalancedTensor.functor (k := k) (R := R) (S := R) (F := F)).mapHomologicalComplex _).obj P
variable (P Q : CochainComplex (ModuleCat.{u} R) ℤ) (n : ℤ)
def crossCochainEquiv : Cochain P (tensorCochain (k := k) (F := F) Q) n ≃+
Cochain (jCochain (k := k) (F := F) P) (eCochain (k := k) (F := F) Q) n where
toFun x := Cochain.mk fun p q hpq => crossHomAdd (P.X p) (Q.X q) (x.v p q hpq)
invFun x := Cochain.mk fun p q hpq => ModuleCat.ofHom (TriangularModule.bottomMap (x.v p q hpq).hom)
left_inv x := by
apply Cochain.ext
intro p q hpq
rfl
right_inv x := by
apply Cochain.ext
intro p q hpq
apply ModuleCat.hom_ext
exact jHom_bottomMap _
map_add' x y := by
apply Cochain.ext
intro p q hpq
exact map_add (crossHomAdd (P.X p) (Q.X q)) (x.v p q hpq) (y.v p q hpq)
@[simp] theorem crossCochainEquiv_v
(x : Cochain P (tensorCochain (k := k) (F := F) Q) n) (p q : ℤ) (hpq : p+n=q) :
(crossCochainEquiv P Q n x).v p q hpq = crossHomAdd (P.X p) (Q.X q) (x.v p q hpq) := rfl
theorem crossCochainEquiv_differential (m : ℤ)
(x : Cochain P (tensorCochain (k := k) (F := F) Q) n) :
δ n m (crossCochainEquiv P Q n x) = crossCochainEquiv P Q m (δ n m x) := by
by_cases hnm : n+1=m
· apply Cochain.ext
intro p q hpq
simp only [δ_v n m hnm _ p q hpq (q-1) (p+1) rfl rfl, crossCochainEquiv_v]
let x₁ : P.X p ⟶ (BalancedTensor.functor (k := k) (R := R) (S := R) (F := F)).obj (Q.X (q-1)) :=
x.v p (q-1) (by omega)
let x₂ : P.X (p+1) ⟶ (BalancedTensor.functor (k := k) (R := R) (S := R) (F := F)).obj (Q.X q) :=
x.v (p+1) q (by omega)
change crossHomAdd _ _ x₁ ≫ (eFunctor k R F).map (Q.d (q-1) q) +
m.negOnePow • ((jFunctor k R F).map (P.d p (p+1)) ≫ crossHomAdd _ _ x₂) =
crossHomAdd (P.X p) (Q.X q) (x₁ ≫
(BalancedTensor.functor (k := k) (R := R) (S := R) (F := F)).map (Q.d (q-1) q) +
m.negOnePow • (P.d p (p+1) ≫ x₂))
rw [map_add]
simp only [Units.smul_def, map_zsmul]
rw [crossHomAdd_target, crossHomAdd_source]
· rw [δ_shape n m hnm, δ_shape n m hnm, map_zero]
def crossFullHomEquivalence : FullHomEquivalence P (tensorCochain (k := k) (F := F) Q)
(jCochain (k := k) (F := F) P) (eCochain (k := k) (F := F) Q) where
cochain := crossCochainEquiv P Q
differential := crossCochainEquiv_differential P Q
def crossCohomologyEquiv :
CohomologyClass P (tensorCochain (k := k) (F := F) Q) n ≃+
CohomologyClass (jCochain (k := k) (F := F) P) (eCochain (k := k) (F := F) Q) n :=
(crossFullHomEquivalence P Q).cohomology n
variable {P Q n}
def crossChainMap (f : P ⟶ tensorCochain (k := k) (F := F) Q) :
jCochain (k := k) (F := F) P ⟶ eCochain (k := k) (F := F) Q where
f i := crossHomAdd (P.X i) (Q.X i) (f.f i)
comm' i j _ := by
change crossHomAdd _ _ (f.f i) ≫ (eFunctor k R F).map (Q.d i j) =
(jFunctor k R F).map (P.d i j) ≫ crossHomAdd _ _ (f.f j)
exact (crossHomAdd_target (f.f i) (Q.d i j)).symm.trans
((congrArg (crossHomAdd (P.X i) (Q.X j)) (f.comm i j)).trans
(crossHomAdd_source (P.d i j) (f.f j)))
theorem crossChainMap_ofHom (f : P ⟶ tensorCochain (k := k) (F := F) Q) :
crossCochainEquiv P Q 0 (Cochain.ofHom f) = Cochain.ofHom (crossChainMap f) := by
apply Cochain.ext₀
intro p
simp only [crossCochainEquiv_v, Cochain.ofHom_v, crossChainMap]
variable {L : CochainComplex (ModuleCat.{u} R) ℤ} {m : ℤ}
theorem crossCochainEquiv_source (a : Cochain P Q n)
(b : Cochain Q (tensorCochain (k := k) (F := F) L) m) :
crossCochainEquiv P L (n+m) (a.comp b rfl) =
(a.map (jFunctor k R F)).comp (crossCochainEquiv Q L m b) rfl := by
apply Cochain.ext
intro p q hpq
simp only [crossCochainEquiv_v, Cochain.comp_v a b rfl p (p+n) q rfl (by omega),
Cochain.comp_v (a.map (jFunctor k R F)) (crossCochainEquiv Q L m b) rfl
p (p+n) q rfl (by omega), Cochain.map_v]
exact crossHomAdd_source _ _
theorem crossCochainEquiv_target
(a : Cochain P (tensorCochain (k := k) (F := F) Q) n) (b : Cochain Q L m) :
crossCochainEquiv P L (n+m)
(a.comp (b.map (BalancedTensor.functor (k := k) (R := R) (S := R) (F := F))) rfl) =
(crossCochainEquiv P Q n a).comp (b.map (eFunctor k R F)) rfl := by
apply Cochain.ext
intro p q hpq
simp only [crossCochainEquiv_v,
Cochain.comp_v a (b.map (BalancedTensor.functor (k := k) (R := R) (S := R) (F := F))) rfl
p (p+n) q rfl (by omega),
Cochain.comp_v (crossCochainEquiv P Q n a) (b.map (eFunctor k R F)) rfl
p (p+n) q rfl (by omega), Cochain.map_v]
exact crossHomAdd_target _ _
end ArExplicit.TriangularInduction
end
end OAI
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