Arc closures and the local isomorphism problem

Abstract

We give an answer in the "geometric" setting to a question of de Fernex, Ein, and Ishii, 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 $\mathfrak a\mapsto \mathfrak a^{\mathrm{ac}}$ (the arc-closure) on a local $k$-algebra $(R,\mathfrak m,L)$ is trivial when $R$ is Noetherian and $k\hookrightarrow L$ is separable, and thus that such a germ $\mathrm{Spec}\ R$ has the (embedded) local isomorphism property.