We consider injective local maps from a local domain R to a local domain S such that the generic fiber of the inclusion map R -\u3e S is trivial, that is P R (0) for every nonzero prime ideal P of S. We present several examples of injective local maps involving power series that have or fail to have this property. For an extension R -\u3e S having this property, we give some results on the dimension of S; in some cases we show dim S = 2 and in some cases dim S = 1
