  This paper studies the behavior of Grobner bases with respect to extensions
of scalars. We prove that an extension of scalars commutes with taking Grobner
bases iff the extension is flat. We consider what information can be deduced
about fibers of a family, from the Grobner basis of the defining ideal of the
family itself. This information can be used to construct algorithms, such as to
find locii over which a map is finite, or an isomorphism.
