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.

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