\begin{definition}[Definition:Morphism of Schemes Locally of Finite Type]
Let $\struct {X, \OO_X}$ and $\struct {Y, \OO_Y}$ be schemes.
Let $f : \struct {X, \OO_X} \to \struct {Y, \OO_Y}$ be a morphism of schemes.
$f$ is '''locally of finite type''' {{iff}} for all affine open subsets $U \subset Y$, there is an open cover $\family {V_i}_{i \mathop \in I}$ of $\map {f^{-1} } U$ by affine open subsets $V_i \subset \map {f^{-1} } U$, such that the ring homomorphism $\map {\OO_Y} U \to \map {\OO_X} {V_i}$ makes $\map {\OO_X} {V_i}$ a finitely generated commutative $\map {\OO_Y} U$-algebra.
\end{definition}