Commutative group scheme
A group scheme over a basis scheme
, the value of which on any
-scheme is an Abelian group. Examples of commutative group schemes are Abelian schemes and algebraic tori (cf. Algebraic torus; 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
there is an open neighbourhood
and and an absolutely-flat quasi-compact morphism
such that the commutative group scheme
is diagonalizable over
. Here, a diagonalizable group scheme is a group scheme of the form
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where is an Abelian group and
is its group algebra with coefficients in the structure sheaf
of the scheme
. In the case when
is the spectrum of an algebraically closed field, this notion reduces to that of a diagonalizable group. If
is the additive group of integers, then
coincides with the multiplicative group scheme
.
Let be a group scheme over
whose fibre over the point
is a group scheme of multiplicative type over the residue class field
. Then there is a neighbourhood
of
such that
is a group scheme of multiplicative type over
(Grothendieck's rigidity theorem).
The structure of commutative group schemes has been studied in the case when the basis scheme is the spectrum of a field
, and the commutative group scheme
is of finite type over
. 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
of such a type has a maximal invariant group subscheme
of multiplicative type, the quotient with respect to which is a unipotent group. If the field
is perfect, then
, where
is a maximal unipotent subgroup of
.
References
[1] | J.-P. Serre, "Groupes algébrique et corps des classes" , Hermann (1959) |
[2] | M. Demazure, A. Grothendieck, "Schémas en groupes II" , Lect. notes in math. , 152 , Springer (1970) |
[3] | M. Demazure, P. Gabriel, "Groupes algébriques" , 1 , Masson (1970) |
[4] | F. Oort, "Commutative group schemes" , Lect. notes in math. , 15 , Springer (1966) |
[5] | W. Waterhouse, "Introduction to affine group schemes" , Springer (1979) |
Comments
A group scheme over a scheme
is an
-scheme such that
is a group for any
-scheme
. If
is an Abelian, or commutative, group for all such
, then
is called a commutative group scheme.
The multiplicative group scheme takes the value
, the group of invertible elements of the ring of functions on
for each
-scheme
. The additive group scheme
takes the values
, the underlying additive group of
. A group scheme over
can equivalently be defined as a group object in the category of
-schemes.
References
[a1] | 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=13667