Finite Groups over Perfect Fields

Theorem. Let G be a finite group scheme over a perfect field. Then G is the semi-direct product of G° and n0 G. 

Corollary. A finite abelian group scheme over a perfect field splits canonically into four factors of the following types ... 

In a natural way D(G) is a 1 A-module, and D is a functor. CARTIER and GABRIEL prove that D is an anti-equivalence between the category of finitely generated A-modules annihilated by some power of V, and the category of unipotent algebraic group schemes over k (k is sup josed to be a perfect field in their theory). Examples ... 

Since the morphism Xo Spec(Fp) is of finite type, the completions of the local rings of Xo at its closed points are Noetherian complete local rings with perfect residue fields. Over these rings we can find a Barsotti-Tate group H with H pk] Go restricted to these rings (see [IL] Theorem 4.4). Hence it follows from Proposition 3.4 that Mx(Go) is locally free over Ox/pkOx of rank h — height(G) at all the closed points of X. ... 

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