The class of Lemma 102 is closed under étale pull-backs covering the cosupport

Kollár's functoriality step for order reduction ([Kol07, 104, Step 2.3]) applies Theorem 92 to the triple handed to BD_{n,m,0} and Theorem 97 to the two resulting sequences; functor_independent_of_maximalContact_of_closed (Hironaka/Resolution/Algebraic/MaximalContact/FunctorIndependence.lean) packages both, for a functor on a class Dom closed under the étale pull-backs whose image contains the cosupport (ClosedUnderEtalePullbackOverCosupp). For Lemma 102's class BDClass n m j this closure is elementary: the relative dimension is unchanged along an étale morphism (Mathlib's smoothOfRelativeDimension_comp with the étale morphism of relative dimension 0), max-ord does not increase under smooth pull-back (maxOrd_comap_le_of_smooth, Hironaka/Resolution/Algebraic/OrderReduction/Functorial.lean), and the inverse image of a divisor family keeps its index set. Used in Hironaka/Resolution/Algebraic/OrderReduction/Step22Indep.lean.

theorem Hironaka.BO.bdClass_closedUnderEtalePullbackOverCosupp {k : Type u} [Field k] [CharZero k] (n m j : ℕ) : AlgebraicGeometry.Scheme.IdealSheafData.ClosedUnderEtalePullbackOverCosupp (AlgebraicGeometry.Triple.BDClass n m j) m

BDClass n m j is closed under pull-back along étale morphisms whose image contains the cosupport: the closure hypothesis of functor_independent_of_maximalContact_of_closed for the class of [Kol07, Lemma 102].