| import OAI.Algebra.AuslanderReiten.Triangular.TriangularCategory |
| import OAI.Algebra.AuslanderReiten.Complete.CompleteShiftDual |
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|
| namespace OAI |
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| noncomputable section |
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|
| namespace ArExplicit.TriangularInduction |
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|
| open CategoryTheory |
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| universe u |
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| 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] |
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| attribute [local instance] moduleCatBaseModule moduleCatBaseTower |
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|
| abbrev eCochain (P : CochainComplex (ModuleCat.{u} R) ℤ) := |
| ((eFunctor k R F).mapHomologicalComplex _).obj P |
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| abbrev jCochain (P : CochainComplex (ModuleCat.{u} R) ℤ) := |
| ((jFunctor k R F).mapHomologicalComplex _).obj P |
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| variable [Module.Finite k R] [Module.Finite k F] |
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| theorem eCochain_totallyAcyclic [Module.Finite Rᵐᵒᵖ F] [Module.Projective Rᵐᵒᵖ F] |
| (P : CochainComplex (ModuleCat.{u} R) ℤ) (hP : TotallyAcyclicCochain P) : |
| TotallyAcyclicCochain (eCochain (k := k) (F := F) P) where |
| finite n := by |
| let := hP.finite n |
| let : Module.Finite k (P.X n) := Module.Finite.trans R (P.X n) |
| exact Module.Finite.of_restrictScalars_finite k (Triangular R F) (TriangularE k R F (P.X n)) |
| projective n := by |
| let := hP.finite n |
| let := hP.projective n |
| exact eProjective |
| exactAt n := by |
| rw [HomologicalComplex.exactAt_iff' _ (n - 1) n (n + 1) (by simp) (by simp), |
| ShortComplex.moduleCat_exact_iff_range_eq_ker] |
| have h := hP.exactAt n |
| rw [HomologicalComplex.exactAt_iff' _ (n - 1) n (n + 1) (by simp) (by simp), |
| ShortComplex.moduleCat_exact_iff_range_eq_ker] at h |
| exact (LinearMap.exact_iff.mp (eMap_exact (k := k) (F := F) _ _ |
| (LinearMap.exact_iff.mpr h.symm))).symm |
| dualExact n := eHomExact _ _ (hP.dualExact n) |
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| omit [Module.Finite k F] in |
| theorem jCochain_totallyAcyclic [Module.Finite R F] [Module.Projective R F] |
| (P : CochainComplex (ModuleCat.{u} R) ℤ) (hP : TotallyAcyclicCochain P) : |
| TotallyAcyclicCochain (jCochain (k := k) (F := F) P) where |
| finite n := by |
| let := hP.finite n |
| let : Module.Finite k (P.X n) := Module.Finite.trans R (P.X n) |
| exact Module.Finite.of_restrictScalars_finite k (Triangular R F) (TriangularJ k R F (P.X n)) |
| projective n := by |
| let := hP.finite n |
| let := hP.projective n |
| exact jProjective |
| exactAt n := by |
| rw [HomologicalComplex.exactAt_iff' _ (n - 1) n (n + 1) (by simp) (by simp), |
| ShortComplex.moduleCat_exact_iff_range_eq_ker] |
| have h := hP.exactAt n |
| rw [HomologicalComplex.exactAt_iff' _ (n - 1) n (n + 1) (by simp) (by simp), |
| ShortComplex.moduleCat_exact_iff_range_eq_ker] at h |
| exact (LinearMap.exact_iff.mp (jMap_exact (k := k) (F := F) _ _ |
| (LinearMap.exact_iff.mpr h.symm))).symm |
| dualExact n := jHomExact _ _ (hP.dualExact n) |
|
|
| end ArExplicit.TriangularInduction |
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| end |
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| end OAI |
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