File size: 3,047 Bytes
216fb31 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 | 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
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