Definition 29.44.1. Let $f : X \to S$ be a morphism of schemes.

We say that $f$ is

*integral*if $f$ is affine and if for every affine open $\mathop{\mathrm{Spec}}(R) = V \subset S$ with inverse image $\mathop{\mathrm{Spec}}(A) = f^{-1}(V) \subset X$ the associated ring map $R \to A$ is integral.We say that $f$ is

*finite*if $f$ is affine and if for every affine open $\mathop{\mathrm{Spec}}(R) = V \subset S$ with inverse image $\mathop{\mathrm{Spec}}(A) = f^{-1}(V) \subset X$ the associated ring map $R \to A$ is finite.

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