Groups of Multiplicative Type

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 ... 

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

This helps indicate why groups of multiplicative type are important. But it should be said that solvability is definitely a necessary hypothesis. Let S for example be the group of all rotations of real 3-space. For g in S we have gtf — 1, so all complex eigenvalues of g have absolute value 1. The characteristic equation of g has odd degree and hence has at least one real root. Since det( ) = 1, it is easy to see that 1 is an eigenvalue. In other words, each rotation leaves a line fixed, and thus it is simply a rotation in the plane perpendicular to that axis (Euler s theorem). Each such rotation is clearly separable. But obviously the group is not commutative (and not solvable). Finally, since U is nilpotent, we have the following result. 

In characteristic p all this fails representations decompose into irreducibles only for groups of multiplicative type. For reductive G one can however prove the following geometric reductivity , which fortunately is enough for many purposes. Suppose G acts linearly on V and 0 v in V is fixed. Then there is a G-invariant homogeneous polynomial function/on V... 

Theorem. An abelian matrix group H consists of separable matrices iff the group scheme G corresponding to H is of multiplicative type. IfH is connected, G is a torus... 

Theorem. Taking character groups yields an anti-equivalence between group schemes of multiplicative type and abelian groups on which 8 = Gal(k, jk) acts continuously. 

Corollary. An algebraic group scheme of multiplicative type is diagonalizable over a finite Galois extension. 

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

Theorem. Let G be a connected affine group scheme acting as automorphisms of an algebraic group scheme T of multiplicative type. Then G acts trivially. 

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

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. 

The construction of A, commutes with base extension, since n0(CD L) = n0(CD) L. Hence to prove G, is of multiplicative type and G unipotent we may assume k = k. Then each CD/Rad CD is a product of copies of k, and the homomorphisms to k are group-like elements spanning C,. Thus A, is spanned by group-likes, and G, is diagonalizable. Also, any group-like b in C defines a homomorphism CD - k such a homomorphism vanishes on the radical, so b is in C,. Thus the other tensor factor of A, representing G , has no nontrivial group-likes. Hence by the previous corollary G is unipotent. ... 

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