Definition 37.76.1. A morphism $f : X \to Y$ of schemes is said to be completely decomposed1 if for all points $y \in Y$ there is a point $x \in X$ with $f(x) = y$ such that the field extension $\kappa (x)/\kappa (y)$ is trivial. A family of morphisms $\{ f_ i : X_ i \to Y\} _{i \in I}$ of schemes with fixed target is said to be completely decomposed if $\coprod f_ i : \coprod Y_ i \to X$ is completely decomposed.

[1] This may be nonstandard terminology.

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