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A group scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023400/c0234001.png" /> over a basis scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023400/c0234002.png" />, the value of which on any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023400/c0234003.png" />-scheme is an Abelian group. Examples of commutative group schemes are Abelian schemes and algebraic tori (cf. [[Algebraic torus|Algebraic torus]]; [[Abelian scheme|Abelian scheme]]). A generalization of algebraic tori in the framework of the theory of group schemes is the following notion. A commutative group scheme is said to be a group scheme of multiplicative type if for any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023400/c0234004.png" /> there is an open neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023400/c0234005.png" /> and and an absolutely-flat quasi-compact morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023400/c0234006.png" /> such that the commutative group scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023400/c0234007.png" /> is diagonalizable over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023400/c0234008.png" />. Here, a diagonalizable group scheme is a group scheme of the form
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{{MSC|14Kxx}}
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{{TEX|done}}
  
<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/c/c023/c023400/c0234009.png" /></td> </tr></table>
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A commutative group scheme is a group scheme $G$ over a basis scheme $S$, the value of which on any
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$S$-scheme is an Abelian group. Examples of commutative group schemes are [[Abelian scheme|Abelian schemes]] and [[Algebraic torus|algebraic tori]]. A generalization of algebraic tori in the framework of the theory of group schemes is the following
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notion. A commutative group scheme is said to be a group scheme of multiplicative type if for any point $s\in S$ there is an open
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neighbourhood $U\ni s$ and and an absolutely-flat quasi-compact morphism $f:U_1\to U$ such that the commutative group scheme $G_1=G\times_U U_1$ is diagonalizable over $U_1$. Here, a diagonalizable group scheme is a group scheme of the form
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$$
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D_S(M) = {\rm Spec}({\mathcal O}_S(M))
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$$ where $M$ is an Abelian group and ${\mathcal O}_S(M)$ is its group algebra with coefficients in the structure sheaf ${\mathcal O}_S$ of the scheme $S$. In the case when $S$ is the spectrum of an algebraically closed field, this notion
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reduces to that of a diagonalizable group. If $M={\mathbb Z}$ is the additive group of integers, then $D_S(M)$ coincides with the multiplicative group scheme $G_{m,S}$.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023400/c02340010.png" /> is an Abelian group and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023400/c02340011.png" /> is its group algebra with coefficients in the structure sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023400/c02340012.png" /> of the scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023400/c02340013.png" />. In the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023400/c02340014.png" /> is the spectrum of an algebraically closed field, this notion reduces to that of a diagonalizable group. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023400/c02340015.png" /> is the additive group of integers, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023400/c02340016.png" /> coincides with the multiplicative group scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023400/c02340017.png" />.
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Let $G$ be a group scheme over $S$ whose fibre over the point $s\in S$ is a group scheme of multiplicative type over the residue class field $k(s)$. Then there is a neighbourhood $U$ of $s$ such that $G\times_S U$ is a group scheme of multiplicative type over $U$ (Grothendieck's rigidity theorem).
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023400/c02340018.png" /> be a group scheme over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023400/c02340019.png" /> whose fibre over the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023400/c02340020.png" /> is a group scheme of multiplicative type over the residue class field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023400/c02340021.png" />. Then there is a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023400/c02340022.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023400/c02340023.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023400/c02340024.png" /> is a group scheme of multiplicative type over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023400/c02340025.png" /> (Grothendieck's rigidity theorem).
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The structure of commutative group schemes has been studied in the case when the basis scheme $S$ is the spectrum of a field $k$, and the commutative group scheme $G$ is of finite type over $k$. In this case the commutative group scheme contains a maximal invariant affine group subscheme, the quotient with respect to which is an Abelian variety (a structure theorem of Chevalley). Any affine commutative group scheme $G$ of such a type has a maximal invariant group subscheme $G_m$ of
 
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multiplicative type, the quotient with respect to which is a unipotent group. If the field $k$ is perfect, then $G\cong G^m\times G^n$, where $G^n$ is a maximal unipotent subgroup of $G$.
The structure of commutative group schemes has been studied in the case when the basis scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023400/c02340026.png" /> is the spectrum of a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023400/c02340027.png" />, and the commutative group scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023400/c02340028.png" /> is of finite type over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023400/c02340029.png" />. In this case the commutative group scheme contains a maximal invariant affine group subscheme, the quotient with respect to which is an Abelian variety (a structure theorem of Chevalley). Any affine commutative group scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023400/c02340030.png" /> of such a type has a maximal invariant group subscheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023400/c02340031.png" /> of multiplicative type, the quotient with respect to which is a unipotent group. If the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023400/c02340032.png" /> is perfect, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023400/c02340033.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023400/c02340034.png" /> is a maximal unipotent subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023400/c02340035.png" />.
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.-P. Serre,   "Groupes algébrique et corps des classes" , Hermann  (1959)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"M. Demazure,   A. Grothendieck,   "Schémas en groupes II" , ''Lect. notes in math.'' , '''152''' , Springer (1970)</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"W. Waterhouse,   "Introduction to affine group schemes" , Springer (1979)</TD></TR></table>
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{|
 
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|-
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|valign="top"|{{Ref|DeGa}}||valign="top"| M. Demazure, P. Gabriel, "Groupes algébriques", '''1''', Masson (1970)
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|-
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|valign="top"|{{Ref|DeGr}}||valign="top"| M. Demazure, A. Grothendieck, "Schémas en groupes II", ''Lect. notes in math.'', '''152''', Springer (1970)
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|-
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|valign="top"|{{Ref|Oo}}||valign="top"| F. Oort, "Commutative group schemes", ''Lect. notes in math.'', '''15''', Springer (1966)
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|-
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|valign="top"|{{Ref|Se}}||valign="top"| J.-P. Serre, "Groupes algébrique et corps des classes", Hermann (1959)
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|-
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|valign="top"|{{Ref|Wa}}||valign="top"| W. Waterhouse, "Introduction to affine group schemes", Springer (1979)
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|-
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|}
  
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====Comments====
  
====Comments====
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A group scheme $G$ over a scheme $S$ is an $S$-scheme such that $G(T)$ is a group for any $S$-scheme $T$. If $G(T)$ is an Abelian, or commutative, group for all such $T$, then $G$ is called a commutative group scheme.
A group scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023400/c02340036.png" /> over a scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023400/c02340037.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023400/c02340038.png" />-scheme such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023400/c02340039.png" /> is a group for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023400/c02340040.png" />-scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023400/c02340041.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023400/c02340042.png" /> is an Abelian, or commutative, group for all such <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023400/c02340043.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023400/c02340044.png" /> is called a commutative group scheme.
 
  
The multiplicative group scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023400/c02340045.png" /> takes the value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023400/c02340046.png" />, the group of invertible elements of the ring of functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023400/c02340047.png" /> for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023400/c02340048.png" />-scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023400/c02340049.png" />. The additive group scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023400/c02340050.png" /> takes the values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023400/c02340051.png" />, the underlying additive group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023400/c02340052.png" />. A group scheme over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023400/c02340053.png" /> can equivalently be defined as a group object in the category of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023400/c02340054.png" />-schemes.
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The multiplicative group scheme $G_{m,S}$ takes the value $\Gamma(T,{\mathcal O}_T)^*$, the group of invertible elements of the ring of functions on $T$ for each $S$-scheme $T$. The additive group scheme $G_{\alpha,S}$ takes the values $G_{\alpha,S}(t) = \Gamma(T,{\mathcal O}_T)^+$, the underlying additive group of $\Gamma(T,{\mathcal O}_T)$. A group scheme over $S$ can equivalently be defined as a group object in the category of $S$-schemes.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"J.-P. Serre,   "Groupes algébrique et corps des classes" , Hermann (1959)</TD></TR></table>
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{|
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|-
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|valign="top"|{{Ref|Se2}}||valign="top"| J.-P. Serre, "Groupes algébrique et corps des classes", Hermann (1959)
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|-
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|}

Latest revision as of 15:38, 26 July 2012

2010 Mathematics Subject Classification: Primary: 14Kxx [MSN][ZBL]

A commutative group scheme is a group scheme $G$ over a basis scheme $S$, the value of which on any $S$-scheme is an Abelian group. Examples of commutative group schemes are Abelian schemes and algebraic tori. A generalization of algebraic tori in the framework of the theory of group schemes is the following notion. A commutative group scheme is said to be a group scheme of multiplicative type if for any point $s\in S$ there is an open neighbourhood $U\ni s$ and and an absolutely-flat quasi-compact morphism $f:U_1\to U$ such that the commutative group scheme $G_1=G\times_U U_1$ is diagonalizable over $U_1$. Here, a diagonalizable group scheme is a group scheme of the form $$ D_S(M) = {\rm Spec}({\mathcal O}_S(M)) $$ where $M$ is an Abelian group and ${\mathcal O}_S(M)$ is its group algebra with coefficients in the structure sheaf ${\mathcal O}_S$ of the scheme $S$. In the case when $S$ is the spectrum of an algebraically closed field, this notion reduces to that of a diagonalizable group. If $M={\mathbb Z}$ is the additive group of integers, then $D_S(M)$ coincides with the multiplicative group scheme $G_{m,S}$.

Let $G$ be a group scheme over $S$ whose fibre over the point $s\in S$ is a group scheme of multiplicative type over the residue class field $k(s)$. Then there is a neighbourhood $U$ of $s$ such that $G\times_S U$ is a group scheme of multiplicative type over $U$ (Grothendieck's rigidity theorem).

The structure of commutative group schemes has been studied in the case when the basis scheme $S$ is the spectrum of a field $k$, and the commutative group scheme $G$ is of finite type over $k$. In this case the commutative group scheme contains a maximal invariant affine group subscheme, the quotient with respect to which is an Abelian variety (a structure theorem of Chevalley). Any affine commutative group scheme $G$ of such a type has a maximal invariant group subscheme $G_m$ of multiplicative type, the quotient with respect to which is a unipotent group. If the field $k$ is perfect, then $G\cong G^m\times G^n$, where $G^n$ is a maximal unipotent subgroup of $G$.

References

[DeGa] M. Demazure, P. Gabriel, "Groupes algébriques", 1, Masson (1970)
[DeGr] M. Demazure, A. Grothendieck, "Schémas en groupes II", Lect. notes in math., 152, Springer (1970)
[Oo] F. Oort, "Commutative group schemes", Lect. notes in math., 15, Springer (1966)
[Se] J.-P. Serre, "Groupes algébrique et corps des classes", Hermann (1959)
[Wa] W. Waterhouse, "Introduction to affine group schemes", Springer (1979)

Comments

A group scheme $G$ over a scheme $S$ is an $S$-scheme such that $G(T)$ is a group for any $S$-scheme $T$. If $G(T)$ is an Abelian, or commutative, group for all such $T$, then $G$ is called a commutative group scheme.

The multiplicative group scheme $G_{m,S}$ takes the value $\Gamma(T,{\mathcal O}_T)^*$, the group of invertible elements of the ring of functions on $T$ for each $S$-scheme $T$. The additive group scheme $G_{\alpha,S}$ takes the values $G_{\alpha,S}(t) = \Gamma(T,{\mathcal O}_T)^+$, the underlying additive group of $\Gamma(T,{\mathcal O}_T)$. A group scheme over $S$ can equivalently be defined as a group object in the category of $S$-schemes.

References

[Se2] J.-P. Serre, "Groupes algébrique et corps des classes", Hermann (1959)
How to Cite This Entry:
Commutative group scheme. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Commutative_group_scheme&oldid=13667
This article was adapted from an original article by I.V. Dolgachev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article