Morphism of finite type

A morphism $f:X\to Y$ is of finite type when every point of $Y$ has an affine neighborhood $V=\mathrm{Spec}A$ for which $f^{-1}V$ has a finite affine cover $\mathrm{Spec}{B}_i$ with each $B_i$ a finitely generated $A$-algebra.

A morphism of schemes $f:X\to S$ is locally of finite type if, over every affine open subscheme $\mathrm{Spec}A\subseteq S$, its inverse image is covered by affine schemes $\mathrm{Spec}B$ with $B$ a finitely generated $A$-algebra. It is of finite type exactly when it is also a quasi-compact morphism.

A finite-type morphism of schemes is quasi-finite if its fibres are finite discrete schemes. A proper quasi-finite morphism is a finite morphism. In particular, a proper finite-type morphism of projective varieties with no positive-dimensional fibres is finite; a positive-dimensional projective fibre would contain an integral curve.