# 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*.

## References

- 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.