English
Same as the Dedekind-domain trace compatibility: the quotient trace matches the localized trace when the base ring is Dedekind and S is integrally closed.
Русский
То же, что и совместимость следа в случае Дедекиндова кольца: факторный след совпадает с локализованным следом.
LaTeX
$$$\\operatorname{trace}_{(R/p)}^{(S/pS)}(\\overline{x}) = \\overline{\\operatorname{trace}_{R}^{S}(x)}$$$
Lean4
instance setLike : SetLike (TwoSidedIdeal R) R
where
coe t := {r | t.ringCon r 0}
coe_injective' := by
rintro ⟨t₁⟩ ⟨t₂⟩ (h : {x | _} = {x | _})
congr 1
refine RingCon.ext fun a b ↦ ⟨fun H ↦ ?_, fun H ↦ ?_⟩
· have H' : a - b ∈ {x | t₁ x 0} := sub_self b ▸ t₁.sub H (t₁.refl b)
rw [h] at H'
convert t₂.add H' (t₂.refl b) using 1 <;> abel
· have H' : a - b ∈ {x | t₂ x 0} := sub_self b ▸ t₂.sub H (t₂.refl b)
rw [← h] at H'
convert t₁.add H' (t₁.refl b) using 1 <;> abel
