fibre of a scheme of finite type over a field being quasi-compact

Question

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!

Answer 1

A scheme is quasicompact iff it can be covered by finitely many open affine subschemes.

When $X, Y$ are affine, it’s easy to see that the fiber is the spectrum of $\mathcal{O}_X(X) \otimes_{\mathcal{O}_Y(Y)} \kappa(y)$ so is quasicompact.

In general, let $y \in V \subset Y$ be an affine open subset, let $f^{-1}(V)=\bigcup_i{U_i}$ be a finite reunion of affine open subsets of $X$ (as $X$ is a Noetherian scheme, all its affine open subsets are quasicompact). Then $X_y$ is the (finite) reunion of the $(U_i)_y$ which are open affine (they are fibers of $U_i \rightarrow V$ affine of finite type), so is quasi-compact.

Answer 2

We know that morphisms of finite type are stable under base extension (see Hartshorne exercise 3.13 (d)). Then $f:X \rightarrow Y$ being of finite type implies that $f^{-1}(y) \rightarrow $ Spec $k(y)$ is of finite type. Note that morphisms of finite type are necessarily quasi-compact (see Hartshorne exercise 3.3(a)), since the preimage of Spec $k(y)$ is $f^{-1}(y)$, we must have it is quasi-compact space.