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About: Radicial morphism     Goto   Sponge   NotDistinct   Permalink

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In algebraic geometry, a morphism of schemes f: X → Y is called radicial or universally injective, if, for every field K the induced map X(K) → Y(K) is injective. (EGA I, (3.5.4)) This is a generalization of the notion of a purely inseparable extension of fields (sometimes called a radicial extension, which should not be confused with a radical extension.) It suffices to check this for K algebraically closed. This is equivalent to the following condition: f is injective on the topological spaces and for every point x in X, the extension of the residue fields k(f(x)) ⊂ k(x)