algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...

, a

morphism In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Al ...

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 In mathematics, an injective function (also known as injection, or one-to-one function ) is a function that maps distinct elements of its domain to distinct elements of its codomain; that is, implies (equivalently by contraposition, impl ...

. (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 field In mathematics, the residue field is a basic construction in commutative algebra. If R is a commutative ring and \mathfrak is a maximal ideal, then the residue field is the quotient ring k=R/\mathfrak, which is a field. Frequently, R is a local ri ...

s :''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

* , section I.3.5. * {{Citation , last1=Bourbaki , first1=Nicolas , author1-link= Nicolas Bourbaki , title=Algebra , publisher=

Springer-Verlag Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in ...

, location=Berlin, New York , isbn=978-3-540-19373-9 , year=1988, see section V.5. Morphisms of schemes