Unipotent Groups

Unipotent groups, unlike groups of multiplicative type, have quite different structure when char(k) 0. The final two sections illustrate this. 

As we have already seen, these results are false in characteristic p explicitly, (o j) xp = x) is an upper triangular copy of Z/pZ. We can however find some restriction on the unipotent groups using Cartier duality. 

Corollaire XI 1.16. Soient S un schema normal noeth rien intfegre, G un S-sch na en groupes lisse sur S, a fibres connexes-, dont la fibre g n rique est une variate ab lienne, et tel que pour tout s e. S de codimension 1, G soit de rang unipotent nul (i.e. G est extension d une variate ab41ienne par un tore). Alors G vdrifie (A). 

As in the last chapter we begin with matrices and then generalize to a class of group schemes the matrices involved here are at the other extreme from separability. What we want is some version of nilpotence, but of course nilpotent matrices cannot occur in a group, so we modify the definition slightly. Call an element g in GL (fc) unipotent if g — 1 is nilpotent— equivalently, all eigenvalues of g should be 1. 

Theorem. Let g be a unipotent element of an algebraic matrix group. Then g acts as a unipotent transformation in every linear representation. Homomar-phisms take unipotent elements to unipotent elements, and unipotence is an intrinsic property. 

Theorem. Let G be a group consisting of unipotent matrices. Then in some basis all elements of G are strictly upper triangular (i.e., zero below the diagonal and 1 on the diagonal). 

Corollary. If a group consists of unipotent matrices, so does its closure. 

Proof. After conjugation, the group will be inside the group U (fc) of all strictly upper triangular matrices. All elements of U (fc) are unipotent, and U (fc) is closed. ... 

The most familiar group of unipotent matrices is U2, which is simply a copy of G. ... 

The last theorem shows us how to define unipotence for arbitrary affine group schemes G is unipotent if every nonzero linear representation has a nonzero fixed vector. For this we must first define fixed vectors, but ob-... 

Corollary., (a) If G is unipotent, so is any closed subgroup and any group scheme represented by a Hopf subalgebra. 

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

Proof. Base-extending to k, we may assume we have a finite constant group scheme, say of order n. When we embed it as an algebraic matrix group, each g in it satisfies the separable equation X" — 1 = 0. If g is also unipotent, g — l. Thus the group is trivial. ... 

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

Corollary. If char(k) = 0, then every unipotent algebraic group scheme is connected. 

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. 

We now begin to study how some more complicated groups are composed of unipotent and multiplicative parts. As usual we start with a theorem on matrices. 

Theorem. Let k be perfect, S an abelian algebraic matrix group. Let Ss and Su be the sets of separable and unipotent elements in S. Then Ss and Su are closed subgroups, and S is their direct product. 

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. 

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