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 $G/G _ {1}$.

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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)