  If a morphism of germs of schemes induces isomorphisms of all local jet
schemes, does it follow that the morphism is an isomorphism? This problem is
called the local isomorphism problem. In this paper, we use jet schemes to
introduce various closure operations among ideals and relate them to the local
isomorphism problem. This approach leads to a partial solution of the local
isomorphism problem, which is shown to have a negative answer in general and a
positive one in several situations of geometric interest.
