  In this paper we study standard bases for submodules of
K[[t_1,...,t_m]][x_1,...,x_n]^s respectively of their localisation with respect
to a t-local monomial ordering. The main step is to prove the existence of a
division with remainder generalising and combining the division theorems of
Grauert and Mora. Everything else then translates naturally. Setting either m=0
or n=0 we get standard bases for polynomial rings respectively for power series
rings as a special case. We then apply this technique to show that the
t-initial ideal of an ideal over the Puiseux series field can be read of from a
standard basis of its generators. This is an important step in the constructive
proof that each point in the tropical variety of such an ideal admits a
lifting.
