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Difference between revisions of "Reduced scheme"

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A [[Scheme|scheme]] whose local ring at any point does not contain non-zero nilpotent elements. For any scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080370/r0803701.png" /> there is a largest closed reduced subscheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080370/r0803702.png" />, characterized by the relations
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A [[Scheme|scheme]] whose local ring at any point does not contain non-zero nilpotent elements. For any scheme $\left({ X,\mathcal{O}_X }\right)$ there is a largest closed reduced subscheme $\left({ X_{\mathrm{red}},\mathcal{O}_{X_{\mathrm{red}}} }\right)$, characterized by the relations
 
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$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080370/r0803703.png" /></td> </tr></table>
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\mathcal{O}_{X_{\mathrm{red}},x} = \mathcal{O}_{X,x}/r_x
 
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$$
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080370/r0803704.png" /> is the ideal of all nilpotent elements of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080370/r0803705.png" />. A [[Group scheme|group scheme]] over a field of characteristic 0 is reduced [[#References|[3]]].
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where $r_x$ is the ideal of all nilpotent elements of the ring $\mathcal{O}_{X,x}$. A [[Group scheme|group scheme]] over a field of characteristic 0 is reduced [[#References|[3]]].
  
 
====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>
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<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>
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<TR><TD valign="top">[2]</TD> <TD valign="top"> A. Grothendieck, J. Dieudonné, "Eléments de géométrie algébrique I. Le langage des schémas" ''Publ. Math. IHES'' , '''4''' (1960) {{MR|0217083}} {{MR|0163908}} {{ZBL|0118.36206}} </TD></TR>
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<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>
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</table>
  
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====Comments====
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That a group scheme over a field of characteristic 0 is reduced is called Cartier's theorem, cf. also [[#References|[a1]]].
  
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It may happen that a scheme $X \rightarrow S$ over a base scheme $S$ is reduced but that $X \times_S T$ is not reduced (even with $S$ and $T$ reduced). The classical objects of study in [[Algebraic geometry|algebraic geometry]] are the algebraic schemes which are reduced and which stay reduced after extending the base field.
  
====Comments====
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====References====
That a group scheme over a field of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080370/r0803706.png" /> is reduced is called Cartier's theorem, cf. also [[#References|[a1]]].
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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>
  
It may happen that a scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080370/r0803707.png" /> over a base scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080370/r0803708.png" /> is reduced but that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080370/r0803709.png" /> is not reduced (even with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080370/r08037010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080370/r08037011.png" /> reduced). The classical objects of study in [[Algebraic geometry|algebraic geometry]] are the algebraic schemes which are reduced and which stay reduced after extending the base field.
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{{TEX|done}}
  
====References====
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[[Category:Algebraic geometry]]
<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>
 

Latest revision as of 07:40, 19 March 2023

A scheme whose local ring at any point does not contain non-zero nilpotent elements. For any scheme $\left({ X,\mathcal{O}_X }\right)$ there is a largest closed reduced subscheme $\left({ X_{\mathrm{red}},\mathcal{O}_{X_{\mathrm{red}}} }\right)$, characterized by the relations $$ \mathcal{O}_{X_{\mathrm{red}},x} = \mathcal{O}_{X,x}/r_x $$ where $r_x$ is the ideal of all nilpotent elements of the ring $\mathcal{O}_{X,x}$. 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 algébrique 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 0 is reduced is called Cartier's theorem, cf. also [a1].

It may happen that a scheme $X \rightarrow S$ over a base scheme $S$ is reduced but that $X \times_S T$ is not reduced (even with $S$ and $T$ 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