Radicial morphism: Difference between revisions

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In algebraic geometry, a morphism 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. (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 fields

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

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 (1): 5–228, doi:10.1007/BF02684778, ISSN 1618-1913, section I.3.5.
Bourbaki, Nicolas (1988), Algebra, Berlin, New York: Springer-Verlag, ISBN 978-3-540-19373-9, see section V.5.

@@ Line 1: / Line 1: @@
-In [[algebraic geometry]], a domain in [[mathematics]], a [[morphism]] of [[scheme (mathematics)|schemes]]
+In [[algebraic geometry]], a [[morphism]] of [[scheme (mathematics)|schemes]]
-:''f'':''X'' &rarr; ''Y''
+:''f'':&nbsp;''X'' → ''Y''
-is called '''radicial''' or '''universally injective''', if, for every ''g'': ''Y' ''&rarr;''Y'' the pullback of ''f'' along ''g'' is [[injective]].
+is called '''radicial''' or '''universally injective''', if, for every field ''K'' the induced map ''X''(''K'')&nbsp;→&nbsp;''Y''(''K'') is [[injective]]. (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: for every point ''x'' in ''X'', the extension of the [[residue field]]s
+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
 :''k''(''f''(''x'')) ⊂ ''k''(''x'')
 is radicial, i.e. [[purely inseparable field extension|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==
-* {{Citation | last1=Grothendieck | first1=Alexandre | author1-link=Alexandre Grothendieck | last2=Dieudonné | first2=Jean | author2-link=Jean Dieudonné | title=Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : I. Le langage des schémas | url=http://www.numdam.org:80/numdam-bin/feuilleter?id=PMIHES_1960__4_ | year=1960 | journal=[[en:Publications Mathématiques de l'IHÉS|Publications Mathématiques de l'IHÉS]] | issn=1618-1913 | volume=4 | pages=5–228}}, section I.3.5.
+* {{Citation | last1=Grothendieck | first1=Alexandre | author1-link=Alexandre Grothendieck | last2=Dieudonné | first2=Jean | author2-link=Jean Dieudonné | title=Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : I. Le langage des schémas | url=http://www.numdam.org:80/numdam-bin/feuilleter?id=PMIHES_1960__4_ | year=1960 | journal=[[Publications Mathématiques de l'IHÉS]] | issn=1618-1913 | volume=4 | issue=1 | pages=5–228 | doi=10.1007/BF02684778}}, section I.3.5.
-* {{Citation | last1=Bourbaki | first1=Nicolas | author1-link=en:Nicolas Bourbaki | title=Algebra | publisher=[[en:Springer-Verlag|Springer-Verlag]] | location=Berlin, New York | isbn=978-3-540-19373-9 | year=1988}}, see section V.5.
+* {{Citation | last1=Bourbaki | first1=Nicolas | author1-link= Nicolas Bourbaki | title=Algebra | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-3-540-19373-9 | year=1988}}, see section V.5.
+[[Category:Morphisms of schemes]]