algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...

, a

morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...

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; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...

. (EGA I, (3.5.4)) This is a generalization of the notion of a

purely inseparable extension In algebra, a purely inseparable extension of fields is an extension ''k'' ⊆ ''K'' of fields of characteristic ''p'' > 0 such that every element of ''K'' is a root of an equation of the form ''x'q'' = ''a'', wit ...

of fields (sometimes called a

radicial extension In algebra, a purely inseparable extension of fields is an extension ''k'' ⊆ ''K'' of fields of characteristic ''p'' > 0 such that every element of ''K'' is a root of an equation of the form ''x'q'' = ''a'', wit ...

, which should not be confused with a

radical extension In mathematics and more specifically in field theory, a radical extension of a field ''K'' is an extension of ''K'' that is obtained by adjoining a sequence of ''n''th roots of elements. Definition A simple radical extension is a simple extensi ...

.) 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 ''m'' is a maximal ideal, then the residue field is the quotient ring ''k'' = ''R''/''m'', which is a field. Frequently, ''R'' is a ...

s :''k''(''f''(''x'')) ⊂ ''k''(''x'') is radicial, i.e.

purely inseparable In algebra, a purely inseparable extension of fields is an extension ''k'' ⊆ ''K'' of fields of characteristic ''p'' > 0 such that every element of ''K'' is a root of an equation of the form ''x'q'' = ''a'', wit ...

. 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