scheme of finite type - geometric interpretation

Question

I'm using the definitions given in Qing Liu's Book:

A morphism $f : X \to Y$ is said to be of finite type if $f$ is quasi-compact, and if for every affine open subset $V$ of $Y$, and for every affine open subset $U$ of $f^{-1}(V)$, the canonical homomorphism $\mathcal O_Y(V) \to \mathcal O_X(U)$ makes $\mathcal O_X(U)$ into a finitely generated $\mathcal O_Y(V)$-algebra. A $Y$-scheme is said to be of finite type if the structural morphism is of finite type.

Now what is the geometric interpretation of a scheme (e.g. a curve or a fibered surface) that is not of finite type? I just see that of finite type seems a nice property that one should have, like for example reduced or separated.

Kind regards, reinbot

Answer 1

Let's decipher a little.

A scheme $X$ over $k$ is of finite type if for every affine $U \subset X$, the morphism $k \to \mathscr O_X(U)$ makes $\mathscr O_X(U)$ into a finitely generated $k$-algebra, but these correspond to affine varieties sitting inside some $\mathbb A^n$.

Thus if a scheme is not of finite type over $k$, it's affine opens cannot be thought of as living inside $\mathbb A^N$, sothey are not very "geometrical".