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 " Bull. Soc. Math. France , 102 (1974) pp. 241–280
[a2]	J.-M. Fontaine, "Il n'y a pas de variété abélienne sur " Invent. Math. , 81 (1985) pp. 515–538
[a3]	G. Faltings, "Endlichkeitssätze für abelsche Varietäten über Zahlkörpern" Invent. Math. , 73 (1983) pp. 349–366 (Erratum: Invent. Math (1984), 381)
[a4]	G. Cornell (ed.) J. Silverman (ed.) , Arithmetic geometry , Springer (1986)