Finite group scheme

A group scheme that is finite and flat over the ground scheme. If $ G $ is a finite group scheme over a scheme $ ( G, {\mathcal O} _ {S} ) $, then $ G = \mathop{\rm Spec} {\mathcal A} $, where $ {\mathcal A} $ is a finite flat quasi-coherent sheaf of algebras over $ {\mathcal O} _ {S} $. From now on it is assumed that $ S $ is locally Noetherian. In this case $ {\mathcal A} $ is locally free. If $ S $ is connected, then the rank of $ {\mathcal A} \times _ { {\mathcal O} _ {s} } k ( s) $ over the field of residues $ k ( s) $ at a point $ s \in S $ is independent of $ s $ and is called the rank of the finite group scheme. Let $ n _ {G} : G \rightarrow G $ be the morphism of $ S $- schemes mapping an element $ s \in G ( T) $ into $ x ^ {n} \in G ( T) $, where $ T $ is an arbitrary $ S $- scheme. The morphism $ n _ {G} $ is null if the rank of $ G $ divides $ n $ and if $ S $ is a reduced scheme or if $ G $ is a commutative finite group scheme (see Commutative group scheme). Every finite group scheme of rank $ p $, where $ p $ is a prime number, is commutative [2].

If $ G _ {1} $ is a subgroup of a finite group scheme $ G $, then one can form the finite group scheme $ G/G _ {1} $, and the rank of $ G $ is the product of the ranks of $ G _ {1} $ and $ G/G _ {1} $.

Examples.

1) Let $ G = G _ {mS} $ be a multiplicative group scheme (or Abelian scheme $ {\mathcal A} $ over $ S $); then $ \mathop{\rm Ker} n _ {G} $ is a finite group scheme of rank $ | n | $( or $ | n | ^ {2 \mathop{\rm dim} {\mathcal A} } $).

2) Let $ S $ be a scheme over the prime field $ \mathbf F _ {p} $ and let $ F: G _ {aS} \rightarrow G _ {aS} $ be the Frobenius homomorphism of the additive group scheme $ G _ {aS} $. Then $ \mathop{\rm Ker} F $ is a finite group scheme of rank $ p $.

3) For every abstract finite group scheme $ \Gamma $ of order $ n $ the constant group scheme $ \Gamma _ {S} $ is a finite group scheme of rank $ n $.

The classification of finite group schemes over arbitrary ground schemes has been achieved in the case where the rank of $ G $ is a prime number (cf. [2]). The case where $ G $ is a commutative finite group scheme and $ S $ is the spectrum of a field of characteristic $ p $ 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 $(p,\ldots,p)$" Bull. Soc. Math. France , 102 (1974) pp. 241–280
[a2]	J.-M. Fontaine, "Il n'y a pas de variété abélienne sur $\ZZ$" 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 75 (1984), 381)
[a4]	G. Cornell (ed.) J. Silverman (ed.) , Arithmetic geometry , Springer (1986)