  In this work we generalize the etching model (Mello et al 2001 Phys. Rev. E
63 041113) to d + 1 dimensions. The dynamic exponents of this model are
compatible with those of the Kardar-Parisi-Zhang universality class. We
investigate the roughness dynamics with surfaces up to d=6. We show that the
data from all substrate lengths and for all dimensions can be collapsed into
one common curve. We determine the dynamic exponents as a function of the
dimension. Moreover, our results suggest that d=4 is not an upper critical
dimension for the etching model, and that it fulfills the Galilean invariance.
