Etale Group Schemes

Theorem. Finite etale group schemes over k are equivalent to finite groups where is acting continuously as group automorphisms. 

Theorem. Let char(k) = 0. Then a nontrivial etale group scheme cannot be unipotent. 

Proof — By definition BrtG = R7i,(BG) where n Sm/Sf, (5m/S) ij is the obvious morphism of sites. Since the third condition of Lemma 3.15 clearly holds for 7t so does the first and therefore it is sufficient to show that BG is A -local in A° ShVet Sm/S). Let, i/9G be a simplicially fibrant model for BG. Using Lemma 2.8(2) wc see that it is sufficient to show that for any striedy henselian local scheme S and a finite etale group scheme G over S of order prime to char S) the map of simplicial sets. G(S). G(As) is a weak equivalence. Since S is strictly henselian G is just a finite group. In particular we obviously have G(S) = G(A ). We also have H],(S,G) = and H],(As, G) = where the second equality holds because of the homotopy invariance of the completion of outside of characteristic ([13]) and therefore our map is a weak equivalence by Proposition 1.16. 

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 an algebraic affine group scheme. Then 7c0(Jc[G]) represents an etale group n0 G, and all maps from G to etale groups factor through the canonical map G - jr0 G. The kernel G° of this map is a connected closed normal subgroup represented by the factor ofk[G] on which s is nonzero. The construction of ic0G and G° commutes with base extension. 

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.]... 

Corollary. Let G be a finite group scheme which is either etale or of multiplicative type. Then Ant(G) is etale. 

Let X be a finite abelian group with 9-action. Associated with X we have a finite etale group (from Chapter 6) and a finite group of multiplicative type (from this chapter). How are these two group schemes related ... 

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

This is actually our second decomposition theorem for abelian groups in (6.8) we decomposed finite abelian group schemes into connected and etale factors. Moreover, that result is of the same type, since by (8.5) we see it is equivalent to a decomposition of the dual into unipotent and multiplicative parts. As this suggests, the theorem in fact holds for all abelian affine group schemes. To introduce the version of duality needed for this extension, we first prove separately a result of some interest in itself. 

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

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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