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.[edit]

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]).

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For some spectacular applications of the results in [a1] see [a2], [a3].

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