Finite group scheme

MODULI OF ABELIAN VARIETIES AND DIEUDONNE MODULES OF FINITE GROUP SCHEMES... 

Moduli of abelian varieties and Dieudonne modules of finite group schemes / Aise Johan de Jong. - [S.l. s.n.]. Proefschrift Nijmegen. - Met lit. opg. - Met samenvatting in het Nederlands. 

Oo] F. Oort, Finite group schemes, local moduli for abelian varieties and lifting problems. In Algebraic Geometry, Oslo, 1970, 223-254. Also in Compositio Math. 23 (1972), 265-296. 

Moduli of abelian varieties and Dieudonne modules of finite group schemes... 

A finite group scheme G over k is called etale if fc[G] is separable. The last theorem shows fc[G] is anti-equivalent to a set X with -action. Also, A k[G] - k[G] k[G] gives a map X x X - X commuting with the <9-action. The dualization here turns the Hopf algebra axioms back into group axioms (see (1.4)). Hence ... 

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

If char(k) = 0, all finite group schemes are in fact etale (11.4), and the other types do not occur. When char(lc) = p, however, we know examples of all four types Z/qZ with q prime to p is etale with etale dual i, while Z/pZ is etale with connected dual pp and vice versa, and p = a is connected with connected dual. The Galois theory of (6.4) describes the first two types, and also (after dualizing) the third. The fourth requires a theory of its own the groups are classified by modules over a certain ring, Dieudonne modules. ... 

Show that a reduced finite group scheme is etale. [G -+ it0 G must be an isomorphism, since G° is trivial and remains so after base extension to ft.]... 

Let G be algebraic of multiplicative type. Show there is a homomorphism from G to a finite group scheme with kernel a torus. 

Let G be a finite group scheme. Show there is a closed embedding of G into the group scheme of units of k[G]°. If G is of multiplicative type, show this embeds G in a torus. 

Let G be a finite group scheme, H any affine group scheme. Show Hom(G, H) is representable. [Embed it in the Weil restriction of H o].]... 

In (11.4) we will show that all finite group schemes in characteristic 0 are etale, and hence none are unipotent. 

Let G be a finite group scheme, not necessarily commutative. Show that G is unipotent iff the augmentation ideal in the (noncommutative) algebra k[G]D is nilpotent. 

Corollary. All finite group schemes in characteristic zero are etale. 

In characteristic p, examples like pp show that the theorem fails in (11. ) we will examine which groups satisfy it. But the first part of the proof still yields some information. We say that a finite group scheme in characteristic p is of height one if xp = 0 for all x in / (this implies connectedness). We can then carry through the lemma with all n less than p. 

A finite group scheme G over arbitrary fc is called etale if Q gi = 0. Show that G is etale if the base-change Gt/n is etale for all maximal ideals Af of k. [See (13.2) and Nakayama s lemma.]... 

Corollary. Let char(k) = p, and let G be a finite group scheme of order prime to p. Then G is etale. 

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