(Quasi-finite)
\(f : X \to Y\) is locally quasi-finite if it is locally of finite type with discrete fibres: every point of every fibre \(X_y = \fiberprod{X}{Y}{\Spec \kappa(y)}\) is isolated, so \(X_y\) is a zero-dimensional scheme. It is quasi-finite if it is in addition quasicompact, equivalently of finite type with finite fibres.
Among finite-type morphisms, finite fibres characterise quasi-finiteness, which is weaker than finiteness. For example, the projection \(V(xy-1)\to\AA^1_k\) onto the \(x\)-coordinate has finite fibres but is not finite: its image \(D(x)\) is not closed.
What is a quasi-finite morphism?
Is a morphism with finite fibres finite?