English
If a property RespectsIso holds, then comparing FormallyUnramified along two equivalent localizations reduces to an equivalence of statements along the localizations; i.e., P(Localization.awayMap f r) holds iff P(IsLocalization.Away.map R' S' f r) holds under suitable isomorphisms.
Русский
Если выполняется свойство RespectsIso, то проверка FormallyUnramified через two эквивалентные локализации сводится к эквивалентности утверждений по локализациям.
LaTeX
$$$\\forall f,r\\ P(\\mathrm{Localization.awayMap}\\ f\\ r) \\iff P(\\mathrm{IsLocalization.Away.map}\\ R'\\ S'\\ f\\ r)$$$
Lean4
/-- A property `RespectsIso` if it still holds when composed with an isomorphism -/
def RespectsIso : Prop :=
(∀ {R S T : Type u} [CommRing R] [CommRing S] [CommRing T],
∀ (f : R →+* S) (e : S ≃+* T) (_ : P f), P (e.toRingHom.comp f)) ∧
∀ {R S T : Type u} [CommRing R] [CommRing S] [CommRing T],
∀ (f : S →+* T) (e : R ≃+* S) (_ : P f), P (f.comp e.toRingHom)
