  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.
