Triviality of arc closures and the local isomorphism problem

Abstract

We give an answer in the "geometric" setting to a question of [dFEI18] asking when local isomorphisms of k-schemes can be detected on the associated maps of local arc or jet schemes. In particular, we show that their ideal-closure operation a → a ac (the arc-closure) on a local k-algebra (R, m, L) is trivial when R is Noetherian and k ֒→ L is separable, and thus that such a germ Spec R has the (embedded) local isomorphism property.