Definition 29.10.1. Let $f : X \to S$ be a morphism.

We say that $f$ is

*universally injective*if and only if for any morphism of schemes $S' \to S$ the base change $f' : X_{S'} \to S'$ is injective (on underlying topological spaces).We say $f$ is

*radicial*if $f$ is injective as a map of topological spaces, and for every $x \in X$ the field extension $\kappa (x) \supset \kappa (f(x))$ is purely inseparable.

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