torsion_by_prime_power_decomposition

English

For a finitely generated p∞-torsion module M over a PID and irreducible p, M decomposes as a direct sum of p^e_i-torsion submodules; i.e., M ≅ ⊕ R/(p_i^{e_i}).

Русский

Для конечнопорожденного p∞-torsion модуля M над ПИД, при ирредуцируемом p mодуля M распадается как прямая сумма p_i^{e_i}-torsion подмодулей: M ≅ ⊕ R/(p_i^{e_i}).

LaTeX

$$$\\text{torsion_by_prime_power_decomposition}\\ R\\ M\\ (hM)\\ h' : \\text{exists } d,k,\\ Nonempty\\ (M \\cong_ℓ[R] \\bigoplus_{i=0}^{d-1} R /(p_i^{k(i)}))$$$

Lean4

/-- A finitely generated `p ^ ∞`-torsion module over a PID is isomorphic to a direct sum of some
  `R ⧸ R ∙ (p ^ e i)` for some `e i`. -/
theorem torsion_by_prime_power_decomposition (hM : Module.IsTorsion' M (Submonoid.powers p)) [h' : Module.Finite R M] :
    ∃ (d : ℕ) (k : Fin d → ℕ), Nonempty <| M ≃ₗ[R] ⨁ i : Fin d, R ⧸ R ∙ p ^ (k i : ℕ) :=
  by
  obtain ⟨d, s, hs⟩ := @Module.Finite.exists_fin _ _ _ _ _ h'; use d; clear h'
  induction d generalizing M with
  | zero =>
    use finZeroElim
    rw [Set.range_eq_empty, Submodule.span_empty] at hs
    haveI : Unique M := ⟨⟨0⟩, fun x => by dsimp; rw [← Submodule.mem_bot R, hs]; exact Submodule.mem_top⟩
    exact ⟨0⟩
  | succ d IH =>
    have : ∀ x : M, Decidable (x = 0) := fun _ => by classical infer_instance
    obtain ⟨j, hj⟩ := exists_isTorsionBy hM d.succ d.succ_ne_zero s hs
    let s' : Fin d → M ⧸ R ∙ s j := Submodule.Quotient.mk ∘ s ∘ j.succAbove
    have := IH ?_ s' ?_
    · obtain ⟨k, ⟨f⟩⟩ := this
      clear IH
      have : ∀ i : Fin d, ∃ x : M, p ^ k i • x = 0 ∧ f (Submodule.Quotient.mk x) = DirectSum.lof R _ _ i 1 :=
        by
        intro i
        let fi := f.symm.toLinearMap.comp (DirectSum.lof _ _ _ i)
        obtain ⟨x, h0, h1⟩ := exists_smul_eq_zero_and_mk_eq hp hM hj fi; refine ⟨x, h0, ?_⟩; rw [h1]
        simp only [fi, LinearMap.coe_comp, f.symm.coe_toLinearMap, f.apply_symm_apply, Function.comp_apply]
      refine ⟨?_, ⟨?_⟩⟩
      · exact fun a => (fun i => (Option.rec (pOrder hM (s j)) k i : ℕ)) (finSuccEquiv d a)
      · refine
          (lequivProdOfRightSplitExact (g := f.toLinearMap.comp <| mkQ _) (f :=
                    (DirectSum.toModule _ _ _ fun i =>
                      (liftQSpanSingleton (p ^ k i) (LinearMap.toSpanSingleton _ _ _) (this i).choose_spec.left :
                        R ⧸ _ →ₗ[R] _)))
                    (R ∙ s j).injective_subtype ?_ ?_).symm ≪≫ₗ
                (((quotTorsionOfEquivSpanSingleton R M (s j)).symm ≪≫ₗ
                      (quotEquivOfEq (torsionOf R M (s j)) _
                        (Ideal.torsionOf_eq_span_pow_pOrder hp hM (s j)))).prodCongr
                  (.refl _ _)) ≪≫ₗ
              (@DirectSum.lequivProdDirectSum R _ _
                  (fun i => R ⧸ R ∙ p ^ @Option.rec _ (fun _ => ℕ) (pOrder hM <| s j) k i) _ _).symm ≪≫ₗ
            (DirectSum.lequivCongrLeft R (finSuccEquiv d).symm)
        · rw [range_subtype, LinearEquiv.ker_comp, ker_mkQ]
        · rw [LinearMap.comp_assoc]
          ext i : 3
          simp only [LinearMap.coe_comp, Function.comp_apply, mkQ_apply]
          rw [LinearEquiv.coe_toLinearMap, LinearMap.id_apply, DirectSum.toModule_lof, liftQSpanSingleton_apply,
            LinearMap.toSpanSingleton_one, Ideal.Quotient.mk_eq_mk, map_one (Ideal.Quotient.mk _),
            (this i).choose_spec.right]
    ·
      exact
        (mk_surjective _).forall.mpr fun x =>
          ⟨(@hM x).choose, by rw [← Quotient.mk_smul, (@hM x).choose_spec, Quotient.mk_zero]⟩
    · have hs' := congr_arg (Submodule.map <| mkQ <| R ∙ s j) hs
      rw [Submodule.map_span, Submodule.map_top, range_mkQ] at hs'; simp only [mkQ_apply] at hs'
      simp only [s']; rw [← Function.comp_assoc, Set.range_comp (_ ∘ s), Fin.range_succAbove]
      rw [← Set.range_comp, ← Set.insert_image_compl_eq_range _ j, Function.comp_apply,
        (Quotient.mk_eq_zero _).mpr (Submodule.mem_span_singleton_self _), Submodule.span_insert_zero] at hs'
      exact hs'

Похожие утверждения