Definition:Morphism of Schemes Locally of Finite Type

Definition

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 if and only if 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.

Sources

• 1977: Robin Hartshorne: Algebraic Geometry $\S \text{II}.3$