| import OAI.Algebra.Finitistic.Tensor.BalancedTensor |
| import Mathlib.RingTheory.Jacobson.Radical |
| import Mathlib.Algebra.Algebra.Opposite |
|
|
| 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 (U : Type u) [AddCommGroup U] [Module k U] [Module Aᵐᵒᵖ U] |
| variable {M N : Type u} [AddCommGroup M] [Module k M] [Module A M] |
| [IsScalarTower k A M] [AddCommGroup N] [Module k N] [Module A N] [IsScalarTower k A N] |
|
|
| def mapRight (f : M →ₗ[A] N) : BalancedTensor.Tensor k A U M →ₗ[k] |
| BalancedTensor.Tensor k A U N := |
| BalancedTensor.lift |
| ((BalancedTensor.mk k A U N).compl₁₂ LinearMap.id (f.restrictScalars k)) (by |
| intro a u m |
| change BalancedTensor.mk k A U N (MulOpposite.op a • u) (f m) = |
| BalancedTensor.mk k A U N u (f (a • m)) |
| rw [f.map_smul, BalancedTensor.balance]) |
|
|
| @[simp] theorem mapRight_mk (f : M →ₗ[A] N) (u : U) (m : M) : |
| mapRight (k := k) U f (BalancedTensor.mk k A U M u m) = |
| BalancedTensor.mk k A U N u (f m) := rfl |
|
|
| omit [Algebra k A] [IsScalarTower k A N] in |
| /-- Radical multiplication is killed by tensor with any right module |
| annihilated by the ring radical. -/ |
| theorem mk_radical_zero |
| (hU : ∀ a ∈ Ring.jacobson A, ∀ u : U, MulOpposite.op a • u = 0) |
| (n : N) (hn : n ∈ Ring.jacobson A • (⊤ : Submodule A N)) (u : U) : |
| BalancedTensor.mk k A U N u n = 0 := by |
| let K : Submodule A N := |
| { carrier := {n | ∀ u : U, BalancedTensor.mk k A U N u n = 0} |
| zero_mem' := fun u => map_zero _ |
| add_mem' := fun hx hy u => by rw [map_add, hx, hy, add_zero] |
| smul_mem' := fun a x hx u => by rw [← BalancedTensor.balance]; exact hx _ } |
| have hK : Ring.jacobson A • (⊤ : Submodule A N) ≤ K := by |
| apply Submodule.smul_le.mpr |
| intro a ha x _ u |
| rw [← BalancedTensor.balance, hU a ha u] |
| exact LinearMap.congr_fun (map_zero (BalancedTensor.mk k A U N)) x |
| exact hK hn u |
|
|
| theorem mapRight_radical_zero |
| (hU : ∀ a ∈ Ring.jacobson A, ∀ u : U, MulOpposite.op a • u = 0) |
| (f : M →ₗ[A] N) |
| (hf : LinearMap.range f ≤ Ring.jacobson A • (⊤ : Submodule A N)) : |
| mapRight (k := k) U f = 0 := by |
| apply BalancedTensor.ext |
| intro u m |
| exact mk_radical_zero (k := k) U hU (f m) (hf (mem_range_self f m)) u |
|
|
| /-- The actual right radical quotient of the regular module. -/ |
| def rightRadical : Submodule Aᵐᵒᵖ A where |
| carrier := Ring.jacobson A |
| zero_mem' := (Ring.jacobson A).zero_mem |
| add_mem' := (Ring.jacobson A).add_mem |
| smul_mem' b _ ha := (Ring.jacobson A).mul_mem_right b.unop ha |
|
|
| abbrev Residue := A ⧸ (rightRadical (A := A)) |
|
|
| theorem residue_annihilated (a : A) (ha : a ∈ Ring.jacobson A) (s : Residue (A := A)) : |
| MulOpposite.op a • s = 0 := by |
| induction s using Submodule.Quotient.induction_on with |
| | _ b => |
| apply (Submodule.Quotient.mk_eq_zero _).mpr |
| exact (Ring.jacobson A).mul_mem_left b ha |
| end LittleFinitistic.PDTensor |
|
|
| end |
|
|
| end OAI |
|
|
