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A group scheme that is finite and flat over the ground scheme. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040290/f0402901.png" /> is a finite group scheme over a scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040290/f0402902.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040290/f0402903.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040290/f0402904.png" /> is a finite flat quasi-coherent sheaf of algebras over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040290/f0402905.png" />. From now on it is assumed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040290/f0402906.png" /> is locally Noetherian. In this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040290/f0402907.png" /> is locally free. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040290/f0402908.png" /> is connected, then the rank of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040290/f0402909.png" /> over the field of residues <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040290/f04029010.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040290/f04029011.png" /> is independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040290/f04029012.png" /> and is called the rank of the finite group scheme. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040290/f04029013.png" /> be the morphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040290/f04029014.png" />-schemes mapping an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040290/f04029015.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040290/f04029016.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040290/f04029017.png" /> is an arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040290/f04029018.png" />-scheme. The morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040290/f04029019.png" /> is null if the rank of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040290/f04029020.png" /> divides <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040290/f04029021.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040290/f04029022.png" /> is a reduced scheme or if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040290/f04029023.png" /> is a commutative finite group scheme (see [[Commutative group scheme|Commutative group scheme]]). Every finite group scheme of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040290/f04029024.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040290/f04029025.png" /> is a prime number, is commutative [[#References|[2]]].
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If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040290/f04029026.png" /> is a subgroup of a finite group scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040290/f04029027.png" />, then one can form the finite group scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040290/f04029028.png" />, and the rank of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040290/f04029029.png" /> is the product of the ranks of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040290/f04029030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040290/f04029031.png" />.
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A group scheme that is finite and flat over the ground scheme. If $  G $
 +
is a finite group scheme over a scheme  $  ( G, {\mathcal O} _ {S} ) $,
 +
then  $  G = \mathop{\rm Spec}  {\mathcal A} $,
 +
where  $  {\mathcal A} $
 +
is a finite flat quasi-coherent sheaf of algebras over  $  {\mathcal O} _ {S} $.  
 +
From now on it is assumed that  $  S $
 +
is locally Noetherian. In this case  $  {\mathcal A} $
 +
is locally free. If  $  S $
 +
is connected, then the rank of  $  {\mathcal A} \times _ { {\mathcal O} _ {s}  } k ( s) $
 +
over the field of residues  $  k ( s) $
 +
at a point  $  s \in S $
 +
is independent of  $  s $
 +
and is called the rank of the finite group scheme. Let  $  n _ {G} : G \rightarrow G $
 +
be the morphism of  $  S $-
 +
schemes mapping an element  $  s \in G ( T) $
 +
into  $  x  ^ {n} \in G ( T) $,
 +
where  $  T $
 +
is an arbitrary  $  S $-
 +
scheme. The morphism  $  n _ {G} $
 +
is null if the rank of  $  G $
 +
divides  $  n $
 +
and if  $  S $
 +
is a reduced scheme or if  $  G $
 +
is a commutative finite group scheme (see [[Commutative group scheme|Commutative group scheme]]). Every finite group scheme of rank  $  p $,
 +
where  $  p $
 +
is a prime number, is commutative [[#References|[2]]].
 +
 
 +
If  $  G _ {1} $
 +
is a subgroup of a finite group scheme  $  G $,  
 +
then one can form the finite group scheme $  G/G _ {1} $,  
 +
and the rank of $  G $
 +
is the product of the ranks of $  G _ {1} $
 +
and $  G/G _ {1} $.
  
 
===Examples.===
 
===Examples.===
  
 +
1) Let  $  G = G _ {mS} $
 +
be a multiplicative group scheme (or Abelian scheme  $  {\mathcal A} $
 +
over  $  S $);
 +
then  $  \mathop{\rm Ker}  n _ {G} $
 +
is a finite group scheme of rank  $  | n | $(
 +
or  $  | n | ^ {2  \mathop{\rm dim}  {\mathcal A} } $).
  
1) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040290/f04029032.png" /> be a multiplicative group scheme (or Abelian scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040290/f04029033.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040290/f04029034.png" />); then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040290/f04029035.png" /> is a finite group scheme of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040290/f04029036.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040290/f04029037.png" />).
+
2) Let $  S $
 +
be a scheme over the prime field  $  \mathbf F _ {p} $
 +
and let  $  F: G _ {aS} \rightarrow G _ {aS} $
 +
be the Frobenius homomorphism of the additive group scheme  $  G _ {aS} $.  
 +
Then  $  \mathop{\rm Ker}  F $
 +
is a finite group scheme of rank $  p $.
  
2) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040290/f04029038.png" /> be a scheme over the prime field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040290/f04029039.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040290/f04029040.png" /> be the Frobenius homomorphism of the additive group scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040290/f04029041.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040290/f04029042.png" /> is a finite group scheme of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040290/f04029043.png" />.
+
3) For every abstract finite group scheme $  \Gamma $
 +
of order  $  n $
 +
the constant group scheme $  \Gamma _ {S} $
 +
is a finite group scheme of rank $  n $.
  
3) For every abstract finite group scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040290/f04029044.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040290/f04029045.png" /> the constant group scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040290/f04029046.png" /> is a finite group scheme of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040290/f04029047.png" />.
+
The classification of finite group schemes over arbitrary ground schemes has been achieved in the case where the rank of $  G $
 
+
is a prime number (cf. [[#References|[2]]]). The case where $  G $
The classification of finite group schemes over arbitrary ground schemes has been achieved in the case where the rank of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040290/f04029048.png" /> is a prime number (cf. [[#References|[2]]]). The case where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040290/f04029049.png" /> is a commutative finite group scheme and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040290/f04029050.png" /> is the spectrum of a field of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040290/f04029051.png" /> is well known (see [[#References|[1]]], [[#References|[3]]], [[#References|[7]]]).
+
is a commutative finite group scheme and $  S $
 +
is the spectrum of a field of characteristic $  p $
 +
is well known (see [[#References|[1]]], [[#References|[3]]], [[#References|[7]]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  Yu.I. Manin,  "The theory of commutative formal groups over fields of finite characteristic"  ''Russian Math. Surveys'' , '''18'''  (1963)  pp. 1–80  ''Uspekhi Mat. Nauk'' , '''18''' :  6  (1963)  pp. 3–90</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J. Tate,  F. Oort,  "Group schemes of prime order"  ''Ann. Sci. Ecole Norm. Sup.'' , '''3'''  (1970)  pp. 1–21</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M. Demazure,  P. Gabriel,  "Groupes algébriques" , '''1''' , Masson  (1970)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  F. Oort,  "Commutative group schemes" , ''Lect. notes in math.'' , '''15''' , Springer  (1966)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  S. Shatz,  "Cohomology of Artinian group schemes over local fields"  ''Ann. of Math.'' , '''79'''  (1964)  pp. 411–449</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  B. Mazur,  "Notes on étale cohomology of number fields"  ''Ann. Sci. Ecole Norm. Sup.'' , '''6'''  (1973)  pp. 521–556</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  H. Kraft,  "Kommutative algebraische Gruppen und Ringe" , Springer  (1975)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  Yu.I. Manin,  "The theory of commutative formal groups over fields of finite characteristic"  ''Russian Math. Surveys'' , '''18'''  (1963)  pp. 1–80  ''Uspekhi Mat. Nauk'' , '''18''' :  6  (1963)  pp. 3–90</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J. Tate,  F. Oort,  "Group schemes of prime order"  ''Ann. Sci. Ecole Norm. Sup.'' , '''3'''  (1970)  pp. 1–21</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M. Demazure,  P. Gabriel,  "Groupes algébriques" , '''1''' , Masson  (1970)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  F. Oort,  "Commutative group schemes" , ''Lect. notes in math.'' , '''15''' , Springer  (1966)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  S. Shatz,  "Cohomology of Artinian group schemes over local fields"  ''Ann. of Math.'' , '''79'''  (1964)  pp. 411–449</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  B. Mazur,  "Notes on étale cohomology of number fields"  ''Ann. Sci. Ecole Norm. Sup.'' , '''6'''  (1973)  pp. 521–556</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  H. Kraft,  "Kommutative algebraische Gruppen und Ringe" , Springer  (1975)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Revision as of 19:39, 5 June 2020


A group scheme that is finite and flat over the ground scheme. If $ G $ is a finite group scheme over a scheme $ ( G, {\mathcal O} _ {S} ) $, then $ G = \mathop{\rm Spec} {\mathcal A} $, where $ {\mathcal A} $ is a finite flat quasi-coherent sheaf of algebras over $ {\mathcal O} _ {S} $. From now on it is assumed that $ S $ is locally Noetherian. In this case $ {\mathcal A} $ is locally free. If $ S $ is connected, then the rank of $ {\mathcal A} \times _ { {\mathcal O} _ {s} } k ( s) $ over the field of residues $ k ( s) $ at a point $ s \in S $ is independent of $ s $ and is called the rank of the finite group scheme. Let $ n _ {G} : G \rightarrow G $ be the morphism of $ S $- schemes mapping an element $ s \in G ( T) $ into $ x ^ {n} \in G ( T) $, where $ T $ is an arbitrary $ S $- scheme. The morphism $ n _ {G} $ is null if the rank of $ G $ divides $ n $ and if $ S $ is a reduced scheme or if $ G $ is a commutative finite group scheme (see Commutative group scheme). Every finite group scheme of rank $ p $, where $ p $ is a prime number, is commutative [2].

If $ G _ {1} $ is a subgroup of a finite group scheme $ G $, then one can form the finite group scheme $ G/G _ {1} $, and the rank of $ G $ is the product of the ranks of $ G _ {1} $ and $ G/G _ {1} $.

Examples.

1) Let $ G = G _ {mS} $ be a multiplicative group scheme (or Abelian scheme $ {\mathcal A} $ over $ S $); then $ \mathop{\rm Ker} n _ {G} $ is a finite group scheme of rank $ | n | $( or $ | n | ^ {2 \mathop{\rm dim} {\mathcal A} } $).

2) Let $ S $ be a scheme over the prime field $ \mathbf F _ {p} $ and let $ F: G _ {aS} \rightarrow G _ {aS} $ be the Frobenius homomorphism of the additive group scheme $ G _ {aS} $. Then $ \mathop{\rm Ker} F $ is a finite group scheme of rank $ p $.

3) For every abstract finite group scheme $ \Gamma $ of order $ n $ the constant group scheme $ \Gamma _ {S} $ is a finite group scheme of rank $ n $.

The classification of finite group schemes over arbitrary ground schemes has been achieved in the case where the rank of $ G $ is a prime number (cf. [2]). The case where $ G $ is a commutative finite group scheme and $ S $ is the spectrum of a field of characteristic $ p $ is well known (see [1], [3], [7]).

References

[1] Yu.I. Manin, "The theory of commutative formal groups over fields of finite characteristic" Russian Math. Surveys , 18 (1963) pp. 1–80 Uspekhi Mat. Nauk , 18 : 6 (1963) pp. 3–90
[2] J. Tate, F. Oort, "Group schemes of prime order" Ann. Sci. Ecole Norm. Sup. , 3 (1970) pp. 1–21
[3] M. Demazure, P. Gabriel, "Groupes algébriques" , 1 , Masson (1970)
[4] F. Oort, "Commutative group schemes" , Lect. notes in math. , 15 , Springer (1966)
[5] S. Shatz, "Cohomology of Artinian group schemes over local fields" Ann. of Math. , 79 (1964) pp. 411–449
[6] B. Mazur, "Notes on étale cohomology of number fields" Ann. Sci. Ecole Norm. Sup. , 6 (1973) pp. 521–556
[7] H. Kraft, "Kommutative algebraische Gruppen und Ringe" , Springer (1975)

Comments

For some spectacular applications of the results in [a1] see [a2], [a3].

References

[a1] M. Raynaud, "Schémas en groupes de type " Bull. Soc. Math. France , 102 (1974) pp. 241–280
[a2] J.-M. Fontaine, "Il n'y a pas de variété abélienne sur " Invent. Math. , 81 (1985) pp. 515–538
[a3] G. Faltings, "Endlichkeitssätze für abelsche Varietäten über Zahlkörpern" Invent. Math. , 73 (1983) pp. 349–366 (Erratum: Invent. Math (1984), 381)
[a4] G. Cornell (ed.) J. Silverman (ed.) , Arithmetic geometry , Springer (1986)
How to Cite This Entry:
Finite group scheme. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Finite_group_scheme&oldid=46929
This article was adapted from an original article by I.V. Dolgachev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article