Radicial morphism
In algebraic geometry, a morphism of schemes
is called radicial or universally injective, if, for every field K the induced map X → Y is injective. This is a generalization of the notion of a purely inseparable extension of fields
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
is radicial, i.e. purely inseparable.
It is also equivalent to every base change of f being injective on the underlying topological spaces.
Radicial morphisms are stable under composition, products and base change. If gf is radicial, so is f.
