| import OAI.Algebra.AuslanderReiten.Triangular.TriangularExact |
| import OAI.Algebra.AuslanderReiten.Triangular.TriangularDualExact |
| import OAI.Algebra.AuslanderReiten.BaseChange.ScalarExtensionComplete |
| import OAI.Algebra.AuslanderReiten.Complete.CompleteComparison |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| namespace ArExplicit.TriangularInduction |
|
|
| open CategoryTheory |
|
|
| universe u |
|
|
| 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] |
|
|
| attribute [local instance] moduleCatBaseModule moduleCatBaseTower |
|
|
| private instance moduleCatBaseComm (M : ModuleCat.{u} R) : SMulCommClass k R M where |
| smul_comm c r m := by |
| change (algebraMap k R c) • (r • m) = r • ((algebraMap k R c) • m) |
| rw [← mul_smul, Algebra.commutes c r, mul_smul] |
|
|
| def eComplex (P : ChainComplex (ModuleCat.{u} R) ℤ) : |
| ChainComplex (ModuleCat.{u} (Triangular R F)) ℤ where |
| X n := ModuleCat.of (Triangular R F) (TriangularE k R F (P.X n)) |
| d i j := ModuleCat.ofHom (eMap (P.d i j).hom) |
| shape i j hij := by |
| rw [P.shape i j hij] |
| apply ModuleCat.hom_ext |
| exact eMap_zero |
| d_comp_d' i j l _ _ := by |
| apply ModuleCat.hom_ext |
| change (eMap (P.d j l).hom).comp (eMap (P.d i j).hom) = 0 |
| rw [← eMap_comp, show (P.d j l).hom.comp (P.d i j).hom = 0 from |
| congrArg ModuleCat.Hom.hom (P.d_comp_d i j l), eMap_zero] |
|
|
| def jComplex (P : ChainComplex (ModuleCat.{u} R) ℤ) : |
| ChainComplex (ModuleCat.{u} (Triangular R F)) ℤ where |
| X n := ModuleCat.of (Triangular R F) (TriangularJ k R F (P.X n)) |
| d i j := ModuleCat.ofHom (jMap (P.d i j).hom) |
| shape i j hij := by |
| rw [P.shape i j hij] |
| apply ModuleCat.hom_ext |
| exact jMap_zero |
| d_comp_d' i j l _ _ := by |
| apply ModuleCat.hom_ext |
| change (jMap (P.d j l).hom).comp (jMap (P.d i j).hom) = 0 |
| rw [← jMap_comp, show (P.d j l).hom.comp (P.d i j).hom = 0 from |
| congrArg ModuleCat.Hom.hom (P.d_comp_d i j l), jMap_zero] |
|
|
| variable [Module.Finite k R] [Module.Finite k F] |
|
|
| theorem eComplex_totallyAcyclic [Module.Finite Rᵐᵒᵖ F] [Module.Projective Rᵐᵒᵖ F] |
| (P : ChainComplex (ModuleCat.{u} R) ℤ) (hP : TotallyAcyclicComplex P) : |
| TotallyAcyclicComplex (eComplex (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 := |
| (LinearMap.exact_iff.mp (eMap_exact (k := k) (F := F) _ _ |
| (LinearMap.exact_iff.mpr (hP.exactAt n).symm))).symm |
| dualExact n := eHomExact _ _ (hP.dualExact n) |
|
|
| omit [Module.Finite k F] in |
| theorem jComplex_totallyAcyclic [Module.Finite R F] [Module.Projective R F] |
| (P : ChainComplex (ModuleCat.{u} R) ℤ) (hP : TotallyAcyclicComplex P) : |
| TotallyAcyclicComplex (jComplex (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 := |
| (LinearMap.exact_iff.mp (jMap_exact (k := k) (F := F) _ _ |
| (LinearMap.exact_iff.mpr (hP.exactAt n).symm))).symm |
| dualExact n := jHomExact _ _ (hP.dualExact n) |
|
|
| end ArExplicit.TriangularInduction |
|
|
| end |
|
|
| end OAI |
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|
