# 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 " 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) |

**How to Cite This Entry:**

Finite group scheme.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Finite_group_scheme&oldid=46929