Definition 29.15.1. Let $f : X \to S$ be a morphism of schemes.

1. We say that $f$ is of finite type at $x \in X$ if there exists an affine open neighbourhood $\mathop{\mathrm{Spec}}(A) = U \subset X$ of $x$ and an affine open $\mathop{\mathrm{Spec}}(R) = V \subset S$ with $f(U) \subset V$ such that the induced ring map $R \to A$ is of finite type.

2. We say that $f$ is locally of finite type if it is of finite type at every point of $X$.

3. We say that $f$ is of finite type if it is locally of finite type and quasi-compact.

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