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

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.We say that $f$ is

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

*finite type*if it is locally of finite type and quasi-compact.

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