| import Mathlib.RingTheory.Finiteness.Projective |
| import Mathlib.RingTheory.Jacobson.Radical |
| import Mathlib.RingTheory.SimpleModule.Basic |
|
|
| namespace OAI |
| noncomputable section |
| namespace LittleFinitistic.PDRadical |
| open Module LinearMap Submodule |
| universe u v |
| variable {A : Type u} [Ring A] |
| variable (P : Type v) [AddCommGroup P] [Module A P] |
|
|
| /-- For a finitely generated projective module the module radical is exactly |
| multiplication by the ring Jacobson radical. No commutativity is assumed. -/ |
| theorem jacobson_eq_smul [Module.Finite A P] [Module.Projective A P] : |
| Module.jacobson A P = Ring.jacobson A • (⊤ : Submodule A P) := by |
| apply le_antisymm _ (Ring.jacobson_smul_top_le A P) |
| intro x hx |
| obtain ⟨n, f, g, hf, hg, hfg⟩ := Module.Finite.exists_comp_eq_id_of_projective A P |
| have hcoord (i : Fin n) : g x i ∈ Ring.jacobson A := |
| Module.le_comap_jacobson ((LinearMap.proj i).comp g) hx |
| have hsum : g x = ∑ i : Fin n, Pi.single i (g x i) := by |
| classical |
| ext j |
| simp |
| have hxsum : x = ∑ i : Fin n, (g x i) • f (Pi.single i 1) := by |
| classical |
| calc |
| x = f (g x) := (DFunLike.congr_fun hfg x).symm |
| _ = ∑ i : Fin n, f (Pi.single i (g x i)) := by conv_lhs => rw [hsum, map_sum] |
| _ = _ := by |
| apply Finset.sum_congr rfl |
| intro i _ |
| have he : Pi.single i (g x i) = (g x i) • (Pi.single i 1 : Fin n → A) := by |
| ext j |
| by_cases hj : i = j |
| · subst j; simp |
| · simp [Ne.symm hj] |
| rw [he, map_smul] |
| rw [hxsum] |
| exact Submodule.sum_mem _ fun i _ => Submodule.smul_mem_smul (hcoord i) mem_top |
|
|
| /-- The noncommutative Nakayama step used in the manuscript. -/ |
| theorem quotient_radical_nontrivial [Module.Finite A P] [Nontrivial P] : |
| Nontrivial (P ⧸ (Ring.jacobson A • (⊤ : Submodule A P))) := by |
| apply Submodule.Quotient.nontrivial_iff.mpr |
| exact ne_of_lt (Submodule.jacobson_smul_lt_top (R := A) (M := P) ⊤) |
|
|
| /-- Any map into a simple module annihilates the radical of its domain. -/ |
| theorem simple_map_radical_zero |
| {S : Type v} [AddCommGroup S] [Module A S] [IsSimpleModule A S] |
| (f : P →ₗ[A] S) (x : P) (hx : x ∈ Module.jacobson A P) : f x = 0 := by |
| have h := Module.le_comap_jacobson f hx |
| have hs : Module.jacobson A S = ⊥ := by |
| apply bot_unique |
| exact sInf_le isCoatom_bot |
| simpa [hs] using h |
| end LittleFinitistic.PDRadical |
|
|
| end |
|
|
| end OAI |
|
|
