The Stacks project

Definition 33.20.1. Let $k$ be a field. An algebraic $k$-scheme is a scheme $X$ over $k$ such that the structure morphism $X \to \mathop{\mathrm{Spec}}(k)$ is of finite type. A locally algebraic $k$-scheme is a scheme $X$ over $k$ such that the structure morphism $X \to \mathop{\mathrm{Spec}}(k)$ is locally of finite type.