Morphism of finite type

Summary

A morphism of finite type $X \to Y$ is a morphism of schemes described by finitely many coordinates over $Y$. Over each affine piece $\operatorname{Spec} B$ of $Y$, the source is covered by finitely many spectra of finitely generated $B$-algebras. It is the relative version of a finitely generated algebra. Varieties over a field $k$ are schemes of finite type over $\operatorname{Spec} k$.