Let $f: X\rightarrow Y$ a smooth morphism over a field $k$. We assume that $X$ is locally of finite type, does it imply that $Y$ is also locally of finite type?

Dear prochet: You've been asking a lot of "basic" algebraic geometry questions recently. Do you have anybody you can speak with about them in person? Anyway, the answer is negative for a silly reason because you omitted the hypothesis that $f$ is surjective (consider $X$ empty, or $Y$ disconnected with $f$ only hitting one connected component, etc.) If $f$ is surjective then the answer is affirmative in much greater generality. See EGA IV$_4$, 17.7.5.
– user61789Jun 30 '13 at 21:18