| import OAI.Algebra.AuslanderReiten.Tensor.BalancedTensorCokernel |
| import OAI.Algebra.AuslanderReiten.Complete.CompleteShiftDual |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace ArExplicit.BalancedTensor |
| open CategoryTheory |
| 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 [Module.Finite S F] [Module.Projective S F] |
| attribute [local instance] moduleCatBaseModule moduleCatBaseTower |
|
|
| theorem complex_exactProjective (P : CochainComplex (ModuleCat.{u} R) ℤ) |
| (hP : ExactProjectiveCochain P) : |
| ExactProjectiveCochain |
| (((functor (k := k) (R := R) (S := S) (F := F)).mapHomologicalComplex _).obj P) where |
| finite n := by |
| let := hP.finite n |
| exact finite |
| projective n := by |
| let := hP.finite n |
| let := hP.projective n |
| exact projective |
| exactAt n := by |
| rw [HomologicalComplex.exactAt_iff' _ (n-1) n (n+1) (by simp) (by simp), |
| ShortComplex.moduleCat_exact_iff_range_eq_ker] |
| have he := hP.exactAt n |
| rw [HomologicalComplex.exactAt_iff' _ (n-1) n (n+1) (by simp) (by simp), |
| ShortComplex.moduleCat_exact_iff_range_eq_ker] at he |
| exact (LinearMap.exact_iff.mp |
| (mapRight_exact (k := k) (F := F) (S := S) _ _ |
| (LinearMap.exact_iff.mpr he.symm))).symm |
|
|
| theorem complex_totallyAcyclic [Module.Injective S S] |
| (P : CochainComplex (ModuleCat.{u} R) ℤ) (hP : ExactProjectiveCochain P) : |
| TotallyAcyclicCochain |
| (((functor (k := k) (R := R) (S := S) (F := F)).mapHomologicalComplex _).obj P) := |
| TotallyAcyclicCochain.of_chain |
| (complex_exactProjective (k := k) (F := F) (S := S) P hP).to_chain.totallyAcyclic |
|
|
| end ArExplicit.BalancedTensor |
|
|
| end |
|
|
| end OAI |
|
|
