  We study the nuclear medium effects on the $c\bar{c}$ time evolution and
charmonium production, in a relativistic proton-nucleus collision. In
particular, we focus on the fragmentation region of the nucleus where the
formation length of the charmonium mesons is shorter than the size of the
nucleus. Little is known on the nuclear effects in this region. We use a
quantum-mechanical model which includes a realistic potential for the
$c\bar{c}$ system and an imaginary potential to describe the collisions of the
$c\bar{c}$ with the nucleons. The imaginary potential introduces a transition
amplitude among the charmonium states and produces an interference pattern on
the charmonium survival probability, which is particulartly important for
$\psi'$. Our results on the suppression factors are compared with data from the
NA50 and E866/NuSea Collaborations. Predictions are given for the suppression
of $J/\psi,~\psi',~\chi_c$ as a function of the nuclear mass and in the
negative $x_F$ region, where data will be available soon.
