Radicial morphism

From Wikipedia, the free encyclopedia

In algebraic geometry, a domain in mathematics, a morphism of schemes

f:X → Y

is called radicial or universally injective, if, for every g: Y' →Y the pullback of f along g is injective.

This is equivalent to the following condition: for every point x in X, the extension of the residue fields

k(f(x)) ⊂ k(x)

is radicial, i.e. purely inseparable.

[edit] 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: 5–228, ISSN 1618-1913, <http://www.numdam.org:80/numdam-bin/feuilleter?id=PMIHES_1960__4_> , section I.3.5.
• Bourbaki, Nicolas (1988), Algebra, Berlin, New York: Springer-Verlag, ISBN 978-3-540-19373-9 , see section V.5.