Unipotent Group Schemes

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. 

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. 

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. 

Theorem. Let G be an abelian affine group scheme over a perfect field. Then G is a product G, x Gu with Gu unipotent and Gt of multiplicative type. 

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 ... 

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