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b797a5b | 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 | 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
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