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Difference between revisions of "Commutative group scheme"

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A group scheme $G$ over a basis scheme $S$, the value of which on any
+
{{MSC|14Kxx}}
$S$-scheme is an Abelian group. Examples of commutative group schemes
+
{{TEX|done}}
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 $s\in S$ 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
+
A commutative group scheme is a group scheme $G$ over a basis scheme $S$, the value of which on any
group scheme of multiplicative type over the residue class field
+
$S$-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
$k(s)$. Then there is a neighbourhood $U$ of $s$ such that $G\times_S U$ is a group
+
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
scheme of multiplicative type over $U$ (Grothendieck's rigidity
+
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
theorem).
+
$$
 +
D_S(M) = {\rm Spec}({\mathcal O}_S(M))
 +
$$ 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 $G_{m,S}$.
  
The structure of commutative group schemes has been studied in the
+
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).
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 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
the commutative group scheme contains a maximal invariant affine group
+
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 $G$.
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====
 
====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"|{{Ref|DeGa}}||valign="top"| M. Demazure, P. Gabriel, "Groupes algébriques", '''1''', Masson (1970)
valign="top"> M. Demazure, A. Grothendieck, "Schémas en groupes II" ,
+
|-
''Lect. notes in math.'' , '''152''' , Springer
+
|valign="top"|{{Ref|DeGr}}||valign="top"| M. Demazure, A. Grothendieck, "Schémas en groupes II", ''Lect. notes in math.'', '''152''', Springer (1970)
(1970)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">
+
|-
M. Demazure, P. Gabriel, "Groupes algébriques" , '''1''' , Masson
+
|valign="top"|{{Ref|Oo}}||valign="top"| F. Oort, "Commutative group schemes", ''Lect. notes in math.'', '''15''', Springer (1966)
(1970)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">
+
|-
F. Oort, "Commutative group schemes" , ''Lect. notes in math.'' ,
+
|valign="top"|{{Ref|Se}}||valign="top"| J.-P. Serre, "Groupes algébrique et corps des classes", Hermann (1959)
'''15''' , Springer (1966)</TD></TR><TR><TD valign="top">[5]</TD> <TD
+
|-
valign="top"> W. Waterhouse, "Introduction to affine group schemes" ,
+
|valign="top"|{{Ref|Wa}}||valign="top"| W. Waterhouse, "Introduction to affine group schemes", Springer (1979)
Springer (1979)</TD></TR></table>
+
|-
 
+
|}
  
 +
====Comments====
  
====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.
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
+
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.
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>
+
|valign="top"|{{Ref|Se2}}||valign="top"| J.-P. Serre, "Groupes algébrique et corps des classes", Hermann (1959)
 +
|-
 +
|}

Latest revision as of 17:38, 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://www.encyclopediaofmath.org/index.php?title=Commutative_group_scheme&oldid=27207
This article was adapted from an original article by I.V. Dolgachev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article