Let $\Bbbk$ be a field and consider the obvious injection $\Bbbk[t]\to \Bbbk(t)$. It seems intuitively obvious this is not a finitely generated $\Bbbk[t]$-algebra, so as a commutative ring morphism it is not of finite type.
Moving over to the geometry it seems being of finite type asserts the existence of a bundle arrow $$\begin{smallmatrix} \mathrm{Spec\;}\Bbbk (t) \\ \downarrow \\ \mathbb A_\Bbbk^1 \end{smallmatrix} \to \begin{smallmatrix} \mathbb A_{\Bbbk [t]}^n \\ \downarrow \\ \mathbb A_\Bbbk^1 \end{smallmatrix}\cong \begin{smallmatrix} \mathbb A_\Bbbk^{n+1} \\ \downarrow \\ \mathbb A_\Bbbk^1 \end{smallmatrix}$$ for some natural $n$. The injection $\Bbbk[t]\to \Bbbk(t)$ seems to correspond to the generic point of the affine line.
What's the geometry here? Why can't there be such a bundle arrow for a natural $n$, and on the other hand why is there always a bundle arrow when $n$ is possibly infinite?