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Difference between revisions of "Reduced scheme"

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====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> M. Artin,   "Algebraic approximation of structures over complete local rings" ''Publ. Math. IHES'' , '''36''' (1969) pp. 23–58</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A. Grothendieck,   J. Dieudonné,   "Eléments de géométrie algebrique I. Le langage des schémas" ''Publ. Math. IHES'' , '''4''' (1960)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> D. Mumford,   "Lectures on curves on an algebraic surface" , Princeton Univ. Press (1966)</TD></TR></table>
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<table><TR><TD valign="top">[1]</TD> <TD valign="top"> M. Artin, "Algebraic approximation of structures over complete local rings" ''Publ. Math. IHES'' , '''36''' (1969) pp. 23–58 {{MR|0268188}} {{ZBL|0181.48802}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A. Grothendieck, J. Dieudonné, "Eléments de géométrie algebrique I. Le langage des schémas" ''Publ. Math. IHES'' , '''4''' (1960) {{MR|0217083}} {{MR|0163908}} {{ZBL|0118.36206}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> D. Mumford, "Lectures on curves on an algebraic surface" , Princeton Univ. Press (1966) {{MR|0209285}} {{ZBL|0187.42701}} </TD></TR></table>
  
  
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> F. Oort,   "Algebraic group schemes in characteristic zero are reduced" ''Invent. Math.'' , '''2''' (1969) pp. 79–80</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> R. Hartshorne,   "Algebraic geometry" , Springer (1977)</TD></TR></table>
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> F. Oort, "Algebraic group schemes in characteristic zero are reduced" ''Invent. Math.'' , '''2''' (1969) pp. 79–80 {{MR|0206005}} {{ZBL|0173.49002}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR></table>

Revision as of 14:51, 24 March 2012

A scheme whose local ring at any point does not contain non-zero nilpotent elements. For any scheme there is a largest closed reduced subscheme , characterized by the relations

where is the ideal of all nilpotent elements of the ring . A group scheme over a field of characteristic 0 is reduced [3].

References

[1] M. Artin, "Algebraic approximation of structures over complete local rings" Publ. Math. IHES , 36 (1969) pp. 23–58 MR0268188 Zbl 0181.48802
[2] A. Grothendieck, J. Dieudonné, "Eléments de géométrie algebrique I. Le langage des schémas" Publ. Math. IHES , 4 (1960) MR0217083 MR0163908 Zbl 0118.36206
[3] D. Mumford, "Lectures on curves on an algebraic surface" , Princeton Univ. Press (1966) MR0209285 Zbl 0187.42701


Comments

That a group scheme over a field of characteristic is reduced is called Cartier's theorem, cf. also [a1].

It may happen that a scheme over a base scheme is reduced but that is not reduced (even with and reduced). The classical objects of study in algebraic geometry are the algebraic schemes which are reduced and which stay reduced after extending the base field.

References

[a1] F. Oort, "Algebraic group schemes in characteristic zero are reduced" Invent. Math. , 2 (1969) pp. 79–80 MR0206005 Zbl 0173.49002
[a2] R. Hartshorne, "Algebraic geometry" , Springer (1977) MR0463157 Zbl 0367.14001
How to Cite This Entry:
Reduced scheme. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Reduced_scheme&oldid=14264
This article was adapted from an original article by S.G. Tankeev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article