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216fb31 | 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 | import OAI.Algebra.Finitistic.Ordinary.OrdinaryTerminal
namespace OAI
noncomputable section
namespace LittleFinitistic.OrdinarySimulation
universe u
variable {k L R O T : Type u} [Field k] [Ring L] [Ring R]
[AddCommGroup O] [AddCommGroup T]
/-- The off-diagonal ordinary bimodule O ⊕ T over L × R. -/
def OffDiagonal (O T : Type u) := O × T
namespace OffDiagonal
instance : AddCommGroup (OffDiagonal O T) := inferInstanceAs (AddCommGroup (O × T))
instance [Module k O] [Module k T] : Module k (OffDiagonal O T) :=
inferInstanceAs (Module k (O × T))
instance leftModule [Module R O] [Module L T] : Module (L × R) (OffDiagonal O T) := by
let : Module (L × R) O := Module.compHom O (RingHom.snd L R)
let : Module (L × R) T := Module.compHom T (RingHom.fst L R)
exact inferInstanceAs (Module (L × R) (O × T))
instance rightModule [Module Lᵐᵒᵖ O] [Module Rᵐᵒᵖ T] :
Module (L × R)ᵐᵒᵖ (OffDiagonal O T) := by
let : Module (L × R)ᵐᵒᵖ O := Module.compHom O (RingHom.fst L R).op
let : Module (L × R)ᵐᵒᵖ T := Module.compHom T (RingHom.snd L R).op
exact inferInstanceAs (Module (L × R)ᵐᵒᵖ (O × T))
instance commuteActions [Module R O] [Module L T] [Module Lᵐᵒᵖ O] [Module Rᵐᵒᵖ T]
[SMulCommClass R Lᵐᵒᵖ O] [SMulCommClass L Rᵐᵒᵖ T] :
SMulCommClass (L × R) (L × R)ᵐᵒᵖ (OffDiagonal O T) where
smul_comm x y z := by
apply Prod.ext
· change x.2 • (MulOpposite.op y.unop.1 • z.1) = MulOpposite.op y.unop.1 • (x.2 • z.1)
exact smul_comm _ _ _
· change x.1 • (MulOpposite.op y.unop.2 • z.2) = MulOpposite.op y.unop.2 • (x.1 • z.2)
exact smul_comm _ _ _
instance towerLeft [Algebra k L] [Algebra k R] [Module k O] [Module k T]
[Module R O] [Module L T] [IsScalarTower k R O] [IsScalarTower k L T] :
IsScalarTower k (L × R) (OffDiagonal O T) where
smul_assoc c x z := Prod.ext (smul_assoc c x.2 z.1) (smul_assoc c x.1 z.2)
instance towerRight [Algebra k L] [Algebra k R] [Module k O] [Module k T]
[Module Lᵐᵒᵖ O] [Module Rᵐᵒᵖ T] [IsScalarTower k Lᵐᵒᵖ O] [IsScalarTower k Rᵐᵒᵖ T] :
IsScalarTower k (L × R)ᵐᵒᵖ (OffDiagonal O T) where
smul_assoc c x z := Prod.ext (smul_assoc c (MulOpposite.op x.unop.1) z.1)
(smul_assoc c (MulOpposite.op x.unop.2) z.2)
instance finiteDimensional [Module k O] [Module k T]
[FiniteDimensional k O] [FiniteDimensional k T] : FiniteDimensional k (OffDiagonal O T) :=
inferInstanceAs (FiniteDimensional k (O × T))
/-- The first left block selects precisely T. -/
theorem first_left [Module R O] [Module L T] (x : OffDiagonal O T) :
((1,0) : L × R) • x = (0,x.2) := by
apply Prod.ext
· exact zero_smul _ _
· exact one_smul _ _
/-- The first right block selects precisely O. -/
theorem first_right [Module Lᵐᵒᵖ O] [Module Rᵐᵒᵖ T] (x : OffDiagonal O T) :
MulOpposite.op ((1,0) : L × R) • x = (x.1,0) := by
apply Prod.ext
· exact one_smul _ _
· exact zero_smul _ _
end OffDiagonal
open CategoryTheory
open scoped TensorProduct
variable {B : Type u} [Ring B] [Algebra k B]
/-- The actual ordinary Section 5 bimodule X over D=B×(B tensor C). -/
abbrev BimoduleX (P : CochainComplex (ModuleCat.{u} (Enveloping k B)) ℤ) (a : ℤ) (l : ℕ) :=
OffDiagonal (BimoduleO P a l) (TerminalTensor B)
theorem ordinary_bimodule (P : CochainComplex (ModuleCat.{u} (Enveloping k B)) ℤ)
(a : ℤ) (l : ℕ) :
SMulCommClass (AlgebraD k l B) (AlgebraD k l B)ᵐᵒᵖ (BimoduleX P a l) := inferInstance
theorem ordinary_finite (P : CochainComplex (ModuleCat.{u} (Enveloping k B)) ℤ)
(a : ℤ) (l : ℕ) [FiniteDimensional k B]
[∀ n, FiniteDimensional k ((underlyingComplex (k := k) P).X n)] :
FiniteDimensional k (BimoduleX P a l) := inferInstance
end LittleFinitistic.OrdinarySimulation
end
end OAI
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