Radicial morphism
Encyclopedia
In algebraic geometry
, a domain in mathematics
, a morphism
of schemes
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))
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
s
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.
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
, a domain in mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a morphism
Morphism
In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures. The notion of morphism recurs in much of contemporary mathematics...
of schemes
Scheme (mathematics)
In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern...
- f: X → Y
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))
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
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...
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.