| import OAI.Algebra.Finitistic.Dimension.PDBalancedRight |
| import OAI.Algebra.Finitistic.Dimension.PDRadical |
|
|
| namespace OAI |
| noncomputable section |
| namespace LittleFinitistic.PDTensor |
| open LinearMap Submodule |
| universe u |
| variable {k A : Type u} [Field k] [Ring A] [Algebra k A] |
| variable (N : Type u) [AddCommGroup N] [Module k N] [Module A N] [IsScalarTower k A N] |
|
|
| abbrev radicalProduct := Ring.jacobson A • (⊤ : Submodule A N) |
| abbrev Top := N ⧸ radicalProduct (A := A) N |
|
|
| def residuePair : Residue (A := A) →ₗ[k] N →ₗ[k] Top (A := A) N := by |
| let f : A →ₗ[k] N →ₗ[k] Top (A := A) N := |
| { toFun := fun a => |
| { toFun := fun n => (radicalProduct N).mkQ (a • n) |
| map_add' := fun x y => by simp [smul_add] |
| map_smul' := fun c n => by simp [smul_comm a c] } |
| map_add' := fun a b => by ext n; simp [add_smul] |
| map_smul' := fun c a => by ext n; simp [smul_assoc] } |
| exact ((rightRadical (A := A)).restrictScalars k).liftQ f (by |
| intro a ha |
| ext n |
| apply (Submodule.Quotient.mk_eq_zero _).mpr |
| exact Submodule.smul_mem_smul ha mem_top) |
|
|
| @[simp] theorem residuePair_mk (a : A) (n : N) : |
| residuePair (k := k) N (Submodule.Quotient.mk a) n = |
| (radicalProduct N).mkQ (a • n) := rfl |
|
|
| def residueEval : BalancedTensor.Tensor k A (Residue (A := A)) N →ₗ[k] Top (A := A) N := |
| BalancedTensor.lift (residuePair (k := k) N) (by |
| intro a s n |
| induction s using Submodule.Quotient.induction_on with |
| | _ b => change (radicalProduct N).mkQ ((b*a) • n) = _; rw [mul_smul]; rfl) |
|
|
| def residueInverse : Top (A := A) N →ₗ[k] BalancedTensor.Tensor k A (Residue (A := A)) N := |
| ((radicalProduct (A := A) N).restrictScalars k).liftQ |
| (BalancedTensor.mk k A (Residue (A := A)) N (Submodule.Quotient.mk (1 : A))) (by |
| intro n hn |
| exact mk_radical_zero (k := k) (Residue (A := A)) residue_annihilated n hn _) |
|
|
| @[simp] theorem residueEval_mk (a : A) (n : N) : |
| residueEval (k := k) N (BalancedTensor.mk k A (Residue (A := A)) N |
| (Submodule.Quotient.mk a) n) = (radicalProduct N).mkQ (a • n) := rfl |
|
|
| @[simp] theorem residueInverse_mk (n : N) : |
| residueInverse (k := k) N ((radicalProduct N).mkQ n) = |
| BalancedTensor.mk k A (Residue (A := A)) N (Submodule.Quotient.mk (1 : A)) n := rfl |
|
|
| /-- The actual noncommutative tensor/quotient identification in cor:pd-detection. -/ |
| def residueTensorEquiv : BalancedTensor.Tensor k A (Residue (A := A)) N ≃ₗ[k] Top (A := A) N where |
| __ := residueEval (k := k) N |
| invFun := residueInverse (k := k) N |
| left_inv z := by |
| change residueInverse (k := k) N (residueEval (k := k) N z) = z |
| induction z using BalancedTensor.induction_on with |
| | hz => simp |
| | ht s n => |
| induction s using Submodule.Quotient.induction_on with |
| | _ a => |
| rw [residueEval_mk, residueInverse_mk, ← BalancedTensor.balance] |
| congr 2 |
| change Submodule.Quotient.mk ((1 : A)*a) = Submodule.Quotient.mk a |
| rw [one_mul] |
| | ha x y hx hy => simp [hx,hy] |
| right_inv z := by |
| induction z using Submodule.Quotient.induction_on with |
| | _ n => |
| change residueEval (k := k) N (residueInverse (k := k) N ((radicalProduct N).mkQ n)) = _ |
| rw [residueInverse_mk, residueEval_mk, one_smul] |
| rfl |
|
|
| /-- Nakayama proves tensor detection for every finitely generated nonzero module. -/ |
| theorem residueTensor_nontrivial [Module.Finite A N] [Nontrivial N] : |
| Nontrivial (BalancedTensor.Tensor k A (Residue (A := A)) N) := by |
| have := PDRadical.quotient_radical_nontrivial (A := A) N |
| exact (residueTensorEquiv (k := k) N).toEquiv.nontrivial |
| end LittleFinitistic.PDTensor |
|
|
| end |
|
|
| end OAI |
|
|
