Radicial morphism

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.)

In algebraic geometry, a morphism of schemes

f: X Y

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)

is radicial, i.e. purely inseparable.

It is also equivalent to every base change of f being injective on the underlying topological spaces. (Thus the term universally injective.)

Radicial morphisms are stable under composition, products and base change. If gf is radicial, so is f.


  • Grothendieck, Alexandre; Dieudonné, Jean (1960), "Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : I. Le langage des schémas", Publications Mathématiques de l'IHÉS, 4 (1): 5–228, doi:10.1007/BF02684778, ISSN 1618-1913, section I.3.5.
  • Bourbaki, Nicolas (1988), Algebra, Berlin, New York: Springer-Verlag, ISBN 978-3-540-19373-9, see section V.5.
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