| import OAI.Algebra.Finitistic.Dimension.PDTor |
| import Mathlib.RingTheory.Jacobson.Ideal |
|
|
| namespace OAI |
| noncomputable section |
| namespace LittleFinitistic.PDTensor |
| open CategoryTheory CategoryTheory.Limits |
| universe u |
| variable {A : Type u} [Ring A] |
|
|
| /-- For an Artinian ring the left-defined ring radical annihilates every |
| actual simple right module. The proof uses cyclicity and a genuine inverse; |
| it does not assume a left/right-radical identification theorem. -/ |
| theorem right_simple_annihilated [IsArtinianRing A] |
| (U : Type u) [AddCommGroup U] [Module Aᵐᵒᵖ U] [IsSimpleModule Aᵐᵒᵖ U] |
| (a : A) (ha : a ∈ Ring.jacobson A) (u : U) : MulOpposite.op a • u=0 := by |
| by_contra hv |
| obtain ⟨b, hb⟩ := IsSimpleModule.toSpanSingleton_surjective Aᵐᵒᵖ hv u |
| change b • (MulOpposite.op a • u) = u at hb |
| have hab : -(a*b.unop) ∈ (⊥ : Ideal A).jacobson := by |
| rw [Ideal.jacobson_bot] |
| exact (Ring.jacobson A).neg_mem ((Ring.jacobson A).mul_mem_right b.unop ha) |
| obtain ⟨z, hz⟩ := Ideal.exists_mul_add_sub_mem_of_mem_jacobson (-(a*b.unop)) hab |
| have hz1 : z * (1-a*b.unop)=1 := by |
| have h : z * (-(a*b.unop)+1)-1=0 := hz |
| simpa [sub_eq_add_neg, add_comm] using sub_eq_zero.mp h |
| have hz2 : (1-a*b.unop)*z=1 := mul_eq_one_comm.mp hz1 |
| have hu : MulOpposite.op (1-a*b.unop) • u=0 := by |
| simp [MulOpposite.op_sub, MulOpposite.op_mul, sub_smul, mul_smul, hb] |
| have hu0 : u=0 := by |
| calc |
| u = MulOpposite.op ((1-a*b.unop)*z) • u := by rw [hz2, MulOpposite.op_one, one_smul] |
| _ = MulOpposite.op z • (MulOpposite.op (1-a*b.unop) • u) := by |
| rw [MulOpposite.op_mul, mul_smul] |
| _ = 0 := by rw [hu, smul_zero] |
| apply hv |
| rw [hu0, smul_zero] |
|
|
| variable {k : Type u} [Field k] [Algebra k A] [FiniteDimensional k A] |
| variable (U : Type u) [AddCommGroup U] [Module k U] [Module Aᵐᵒᵖ U] |
| [IsSimpleModule Aᵐᵒᵖ U] |
| variable (M : PDMinimal.FiniteModule k A) |
|
|
| /-- The derived tensor with every actual simple right module is the tensor |
| of that simple module with the n-th minimal projective-resolution term. -/ |
| def simpleTorIsoTerm (n : ℕ) : Tor (k := k) U M.cat n ≅ |
| (tensorFunctor k A U).obj (PDMinimal.term M n).cat := by |
| haveI : IsArtinianRing A := isArtinian_of_tower k inferInstance |
| exact torIsoTerm U M (right_simple_annihilated U) n |
| end LittleFinitistic.PDTensor |
|
|
| end |
|
|
| end OAI |
|
|
