| 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] |
|
|
| / |
| 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)) |
| |
| / |
| 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 _ _ |
| |
| / |
| 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] |
| |
| / |
| 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 |
| |
