| import Mathlib.LinearAlgebra.Isomorphisms |
| import Mathlib.RingTheory.Ideal.Colon |
| import Mathlib.RingTheory.Ideal.Span |
|
|
| namespace OAI |
|
|
| namespace PiExponentJets.W22 |
|
|
| variable {R : Type*} [CommRing R] |
|
|
| noncomputable def cyclicFactorEquiv (I : Ideal R) (f : R) : |
| (R ⧸ I.colon {f}) ≃ₗ[R] |
| (↥(I ⊔ Ideal.span {f}) ⧸ I.submoduleOf (I ⊔ Ideal.span {f})) := by |
| let J : Ideal R := I ⊔ Ideal.span {f} |
| have hfJ : f ∈ J := (show Ideal.span {f} ≤ J from le_sup_right) |
| (Ideal.subset_span (by simp)) |
| let F : R →ₗ[R] J ⧸ I.submoduleOf J := |
| (I.submoduleOf J).mkQ.comp (LinearMap.toSpanSingleton R J ⟨f, hfJ⟩) |
| have hker : LinearMap.ker F = I.colon {f} := by |
| ext a |
| change ((Submodule.Quotient.mk (a • (⟨f, hfJ⟩ : J)) : J ⧸ I.submoduleOf J) = 0) ↔ |
| a ∈ I.colon {f} |
| rw [Submodule.Quotient.mk_eq_zero] |
| change a * f ∈ I ↔ a ∈ I.colon {f} |
| simp only [Submodule.mem_colon_singleton, smul_eq_mul] |
| have hsurj : Function.Surjective F := by |
| intro y |
| obtain ⟨x, rfl⟩ := (I.submoduleOf J).mkQ_surjective y |
| have hx : (x : R) ∈ Ideal.span {f} ⊔ I := by |
| simpa only [J, sup_comm] using x.property |
| obtain ⟨a, b, hb, hab⟩ := Ideal.mem_span_singleton_sup.mp hx |
| refine ⟨a, ?_⟩ |
| change (Submodule.Quotient.mk (a • (⟨f, hfJ⟩ : J)) : J ⧸ I.submoduleOf J) = |
| Submodule.Quotient.mk x |
| rw [Submodule.Quotient.eq] |
| change a * f - (x : R) ∈ I |
| rw [← hab] |
| simpa only [sub_add_eq_sub_sub, sub_self, zero_sub] using I.neg_mem hb |
| exact (Submodule.quotEquivOfEq _ _ hker.symm).trans (F.quotKerEquivOfSurjective hsurj) |
|
|
| end PiExponentJets.W22 |
|
|
| end OAI |
|
|
