# Finite group scheme

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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 .

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 .

## Contents

### 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. ). The case where is a commutative finite group scheme and is the spectrum of a field of characteristic is well known (see , , ).

How to Cite This Entry:
Finite group scheme. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Finite_group_scheme&oldid=15344
This article was adapted from an original article by I.V. Dolgachev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article