| import OAI.Algebra.Finitistic.Ordinary.OrdinaryBlockModules |
| import OAI.Algebra.Finitistic.Ordinary.OrdinaryProductSplitting |
| import OAI.Algebra.Finitistic.Ordinary.OrdinaryOffDiagonalObjects |
|
|
| namespace OAI |
| noncomputable section |
| open CategoryTheory |
| open scoped TensorProduct |
| namespace LittleFinitistic.BlockModule |
| universe u |
| variable (k L R : Type u) [Field k] [Ring L] [Ring R] [Algebra k L] [Algebra k R] |
|
|
| abbrev firstComplex (Z : CochainComplex (ModuleCat.{u} L) ℤ) := |
| ((ProductModules.firstInflation L R).mapHomologicalComplex (.up ℤ)).obj Z |
| abbrev secondComplex (Z : CochainComplex (ModuleCat.{u} R) ℤ) := |
| ((ProductModules.secondInflation L R).mapHomologicalComplex (.up ℤ)).obj Z |
|
|
| variable {U V : Type u} [AddCommGroup U] [AddCommGroup V] [Module k U] [Module k V] |
|
|
| def firstMap [Module L U] [Module L V] (f : U →ₗ[L] V) : First k L R U →ₗ[L × R] First k L R V where |
| __ := f.toAddHom |
| map_smul' d x := f.map_smul d.1 x |
|
|
| def secondMap [Module R U] [Module R V] (f : U →ₗ[R] V) : Second k L R U →ₗ[L × R] Second k L R V where |
| __ := f.toAddHom |
| map_smul' d x := f.map_smul d.2 x |
| end LittleFinitistic.BlockModule |
| namespace LittleFinitistic.OrdinarySimulation.OffDiagonal |
| universe u |
| variable (k L R : Type u) [Field k] [Ring L] [Ring R] [Algebra k L] [Algebra k R] |
| variable (O : CochainComplex (ModuleCat.{u} (R ⊗[k] Lᵐᵒᵖ)) ℤ) |
| variable (T : CochainComplex (ModuleCat.{u} (L ⊗[k] Rᵐᵒᵖ)) ℤ) |
|
|
| theorem complex_d_all (n m : ℤ) : (complex k L R O T).d n m = objectMap k L R (O.d n m) (T.d n m) := by |
| by_cases h : n+1=m |
| · subst m |
| rw [complex_d] |
| rfl |
| · have hh : ¬ (ComplexShape.up ℤ).Rel n m := h |
| rw [(complex k L R O T).shape n m hh,O.shape n m hh,T.shape n m hh] |
| apply ModuleCat.hom_ext |
| apply LinearMap.ext |
| intro x |
| rfl |
| end LittleFinitistic.OrdinarySimulation.OffDiagonal |
|
|
| end |
|
|
| end OAI |
|
|
