Let \(X = V(xy - 1) \subseteq \AA^2_k\) and let \(f : X \to \AA^1_k\) be the projection to \(x\). Then \(X \cong \GG_m\), every fibre is a single reduced point or empty, so \(f\) is quasi-finite, and the image is \(\AA^1 \sm \ts{0}\), which is not closed. A finite morphism is closed, so \(f\) is not finite, and one sees it on rings: \(k[x] \to k[x, x^{-1}]\) is of finite type and not module-finite.
The same hyperbola also witnesses failure of universal closedness for \(\AA^1\): it is closed in \(\AA^1\times\AA^1\) while its projection is not closed.
The example has two consequences. Zariski’s main theorem repairs it: a separated quasi-finite morphism is an open immersion followed by a finite one, and here the finite morphism is the identity of \(\AA^1\) and the open immersion is \(\GG_m \injects \AA^1\). Over a locally Noetherian base, finite is exactly proper plus quasi-finite, and the failure above is the failure of properness.
Dependencies
D-MORQFQuasi-finite morphismsuses