Let $X,Y$ be schemes of finite type over a field $k$. In particular, they are quasi-compact. Let $f: X \to Y$ be a morphism of finite type (for all open affines $U \subset Y$, $f^{-1}(U)$ is quasi-compact and $\Gamma(V, O\_X)$ is finitely generated over $\Gamma(U, O\_Y)$ for all open affine $V \subset f^{-1}(U)$).

I would like to deduce that the fibre $f^{-1}(y)$ as a scheme over $\operatorname{Spec}\kappa(y)$ is quasi-compact, for any $y \in Y$. How can I prove this? Any comments are appreciated. Thank you!