| import OAI.Algebra.AuslanderReiten.Triangular.TriangularFunctor |
| import Mathlib.Algebra.Module.Projective |
| import Mathlib.RingTheory.Finiteness.Cardinality |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| namespace ArExplicit.TriangularInduction |
|
|
| private instance finitePiProjective {S P ι : Type*} [Ring S] [AddCommGroup P] |
| [Module S P] [Module.Projective S P] [Fintype ι] : Module.Projective S (ι → P) := |
| Module.Projective.of_equiv (DFinsupp.linearEquivFunOnFintype : |
| (Π₀ _ : ι, P) ≃ₗ[S] (ι → P)) |
|
|
| variable {k R F : Type*} [CommRing 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] |
|
|
| def eRegularInclusion : TriangularE k R F R →ₗ[Triangular R F] Triangular R F where |
| toFun x := ⟨x.top, 0, BalancedTensor.runitLeft (S := R) x.bottom⟩ |
| map_add' x y := by ext <;> simp |
| map_smul' a x := by |
| ext |
| · rfl |
| · simp |
| · change BalancedTensor.runitLeft (S := R) |
| (BalancedTensor.tmul a.off x.top + a.bottom • x.bottom) = |
| a.bottom • BalancedTensor.runitLeft (S := R) x.bottom + MulOpposite.op x.top • a.off |
| rw [map_add, map_smul] |
| exact add_comm _ _ |
|
|
| def eRegularProjection : Triangular R F →ₗ[Triangular R F] TriangularE k R F R where |
| toFun a := ⟨a.top, (BalancedTensor.runitLeft (k := k) (R := R) (S := R)).symm a.off⟩ |
| map_add' a b := by ext <;> simp |
| map_smul' a b := by |
| ext |
| · rfl |
| · change (BalancedTensor.runitLeft (S := R)).symm |
| (a.bottom • b.off + MulOpposite.op b.top • a.off) = |
| BalancedTensor.tmul a.off b.top + |
| a.bottom • (BalancedTensor.runitLeft (S := R)).symm b.off |
| apply (BalancedTensor.runitLeft (S := R)).injective |
| rw [LinearEquiv.apply_symm_apply, map_add, map_smul, LinearEquiv.apply_symm_apply] |
| exact add_comm _ _ |
|
|
| omit [IsScalarTower k R F] [SMulCommClass k Rᵐᵒᵖ F] in |
| theorem eRegular_split : |
| (eRegularProjection (k := k) (R := R) (F := F)).comp eRegularInclusion = LinearMap.id := by |
| apply LinearMap.ext |
| intro x |
| ext |
| · rfl |
| · exact (BalancedTensor.runitLeft (S := R)).symm_apply_apply x.bottom |
|
|
| instance eRegular_projective : Module.Projective (Triangular R F) (TriangularE k R F R) := |
| Module.Projective.of_split eRegularInclusion eRegularProjection eRegular_split |
|
|
| def jRegularInclusion : TriangularJ k R F R →ₗ[Triangular R F] Triangular R F where |
| toFun x := ⟨0, x.bottom, 0⟩ |
| map_add' x y := by ext <;> simp |
| map_smul' a x := by ext <;> simp |
| |
| def jRegularProjection : Triangular R F →ₗ[Triangular R F] TriangularJ k R F R where |
| toFun a := ⟨0, a.bottom⟩ |
| map_add' a b := by ext; simp |
| map_smul' a b := by ext; simp |
| |
| omit [Algebra k R] [IsScalarTower k R F] [IsScalarTower k Rᵐᵒᵖ F] |
| [SMulCommClass k Rᵐᵒᵖ F] in |
| theorem jRegular_split : |
| (jRegularProjection (k := k) (R := R) (F := F)).comp jRegularInclusion = LinearMap.id := by |
| apply LinearMap.ext |
| intro x |
| ext |
| exact rfl |
|
|
| instance jRegular_projective : Module.Projective (Triangular R F) (TriangularJ k R F R) := |
| Module.Projective.of_split jRegularInclusion jRegularProjection jRegular_split |
|
|
| variable {M : Type*} [AddCommGroup M] [Module k M] [Module R M] |
| variable [IsScalarTower k R M] [SMulCommClass k R M] |
|
|
| omit [SMulCommClass k R M] in |
| theorem eProjective [Module.Finite R M] [Module.Projective R M] : |
| Module.Projective (Triangular R F) (TriangularE k R F M) := by |
| obtain ⟨n, p, hp⟩ := Module.Finite.exists_fin' R M |
| obtain ⟨s, hs⟩ := Module.projective_lifting_property p (LinearMap.id : M →ₗ[R] M) hp |
| let : Module.Projective (Triangular R F) (TriangularE k R F (Fin n → R)) := |
| Module.Projective.of_equiv (ePi (k := k) (R := R) (F := F) (M := R) (Fin n)).symm |
| apply Module.Projective.of_split (eMap s) (eMap p) |
| rw [← eMap_comp, hs, eMap_id] |
|
|
| omit [SMulCommClass k R M] in |
| theorem jProjective [Module.Finite R M] [Module.Projective R M] : |
| Module.Projective (Triangular R F) (TriangularJ k R F M) := by |
| obtain ⟨n, p, hp⟩ := Module.Finite.exists_fin' R M |
| obtain ⟨s, hs⟩ := Module.projective_lifting_property p (LinearMap.id : M →ₗ[R] M) hp |
| let : Module.Projective (Triangular R F) (TriangularJ k R F (Fin n → R)) := |
| Module.Projective.of_equiv (jPi (k := k) (R := R) (F := F) (M := R) (Fin n)).symm |
| apply Module.Projective.of_split (jMap s) (jMap p) |
| rw [← jMap_comp, hs, jMap_id] |
|
|
| end ArExplicit.TriangularInduction |
|
|
| end |
|
|
| end OAI |
|
|
