I have read that if $f:X\to Y$ is a morphism of projective varieties and $D$ is an ample Cartier divisor on $Y$ then $f^*D$ is nef. The proof is that ample implies semi-ample, which means that some multiple is base-point free. This implies that some multiple of the pullback is also base-point free which implies nef-ness of $f^*D$.
My confusion is over two things - first, where is the projective hypothesis used in this argument? Second, I don't really understand how the pullback of $D$ is defined when $f$ is arbitrary, so I don't really understand how we know that the pullback of a base-point free divisor is base-point free.
If $f$ is flat then we can pullback $D$ by just taking it's inverse image, which will be a Cartier divisor. In this case it seems clear that the pullback of a base-point free divisor would be base-point free, and this does not seem to use the projective assumption unless I am mistaken. So it seems to me that the pullback of an ample divisor by a flat morphism is nef, even without the projective assumption. Is this correct?