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refactor: making trans usage explicit with kerLift #1

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menon-codes merged 1 commit into from
Jul 29, 2026
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refactor: making trans usage explicit with kerLift #1

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8 changes: 3 additions & 5 deletions Mathlib/RingTheory/NoetherNormalization.lean
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Expand Up @@ -258,11 +258,9 @@ theorem exists_integral_inj_algHom_of_quotient (I : Ideal (MvPolynomial (Fin n)
set ϕ := kerLiftAlg <| hom2 f I
have := Quotient.nontrivial_iff.mpr hi
obtain ⟨s, _, g, injg, intg⟩ := hd (ker <| hom2 f I) (ker_ne_top <| hom2 f I)
have comp : (kerLiftAlg (hom2 f I)).comp (Quotient.mkₐ k <| ker <| hom2 f I) = (hom2 f I) :=
AlgHom.ext fun a ↦ by
simp only [AlgHom.coe_comp, Quotient.mkₐ_eq_mk, Function.comp_apply, kerLiftAlg_mk]
exact ⟨s, by lia, ϕ.comp g, (ϕ.coe_comp g) ▸ (kerLiftAlg_injective _).comp injg,
intg.trans _ _ <| (comp ▸ hom2_isIntegral f I fne fi).tower_top _ _⟩
have intgϕ := (hom2_isIntegral f I fne fi).kerLift
exact ⟨s, by lia, ϕ.comp g, (ϕ.coe_comp g) ▸ (kerLiftAlg_injective _).comp injg,
intg.trans g.toRingHom ϕ.toRingHom intgϕ⟩

variable (k R : Type*) [Field k] [CommRing R] [Nontrivial R] [a : Algebra k R]
[fin : Algebra.FiniteType k R]
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