Difference between revisions of "Commutative group scheme"
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− | + | 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 | + | $S$-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 |
− | $ | + | 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 | + | 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 $G_{m,S}$. | ||
− | 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==== | ||
− | + | {| | |
− | valign="top" | + | |- |
− | + | |valign="top"|{{Ref|DeGa}}||valign="top"| M. Demazure, P. Gabriel, "Groupes algébriques", '''1''', Masson (1970) | |
− | valign="top" | + | |- |
− | ''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) | + | |- |
− | + | |valign="top"|{{Ref|Oo}}||valign="top"| F. Oort, "Commutative group schemes", ''Lect. notes in math.'', '''15''', Springer (1966) | |
− | + | |- | |
− | 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) | + | |- |
− | valign="top" | + | |valign="top"|{{Ref|Wa}}||valign="top"| W. Waterhouse, "Introduction to affine group schemes", Springer (1979) |
− | 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. | |
− | 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==== | ||
− | + | {| | |
− | valign="top" | + | |- |
− | Hermann (1959) | + | |valign="top"|{{Ref|Se2}}||valign="top"| J.-P. Serre, "Groupes algébrique et corps des classes", Hermann (1959) |
+ | |- | ||
+ | |} |
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) |
Commutative group scheme. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Commutative_group_scheme&oldid=19578