Finite group scheme
A group scheme that is finite and flat over the ground scheme. If is a finite group scheme over a scheme
, then
, where
is a finite flat quasi-coherent sheaf of algebras over
. From now on it is assumed that
is locally Noetherian. In this case
is locally free. If
is connected, then the rank of
over the field of residues
at a point
is independent of
and is called the rank of the finite group scheme. Let
be the morphism of
-schemes mapping an element
into
, where
is an arbitrary
-scheme. The morphism
is null if the rank of
divides
and if
is a reduced scheme or if
is a commutative finite group scheme (see Commutative group scheme). Every finite group scheme of rank
, where
is a prime number, is commutative [2].
If is a subgroup of a finite group scheme
, then one can form the finite group scheme
, and the rank of
is the product of the ranks of
and
.
Examples.
1) Let be a multiplicative group scheme (or Abelian scheme
over
); then
is a finite group scheme of rank
(or
).
2) Let be a scheme over the prime field
and let
be the Frobenius homomorphism of the additive group scheme
. Then
is a finite group scheme of rank
.
3) For every abstract finite group scheme of order
the constant group scheme
is a finite group scheme of rank
.
The classification of finite group schemes over arbitrary ground schemes has been achieved in the case where the rank of is a prime number (cf. [2]). The case where
is a commutative finite group scheme and
is the spectrum of a field of characteristic
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 ![]() |
[a2] | J.-M. Fontaine, "Il n'y a pas de variété abélienne sur ![]() |
[a3] | G. Faltings, "Endlichkeitssätze für abelsche Varietäten über Zahlkörpern" Invent. Math. , 73 (1983) pp. 349–366 (Erratum: Invent. Math ![]() |
[a4] | G. Cornell (ed.) J. Silverman (ed.) , Arithmetic geometry , Springer (1986) |
Finite group scheme. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Finite_group_scheme&oldid=46929