Diagonalizable Group Schemes

Let G be a diagonalizable group scheme over a field k. Show that every linear representation of G is a direct sum of one-dimensional representations. How are the one-dimensional representations classified ... 

Dense set 157 Deploye, see Split Derivation 83 Derived group 73 Descent data 131 Diagonalizable group scheme 14 Differential field 77 Differential operator 99 Differentials of an algebra 84 Dimension of an algebraic G 88 Direct limit 151... 

Let M be an abelian group, G the diagonalizable group scheme represented by k[M],... 

Theorem. Let M be a subgroup of GL fc). The elements of M can be simuhan-eously diagonalized iff the group scheme G corresponding to M is diagonalizable. 

Separable algebras, besides describing connected components, are related to a familiar kind of matrix and can lead us to another class of group schemes. One calls an n x n matrix g separable if the subalgebra k[p] of End(/c") is separable. We have of course k[g] k[X]/p(X) where p(X) is the minimal polynomial of g. Separability then holds iff k[g] k = /qg] a fc(Y]/p(.Y) is separable over k. This means that p has no repeated roots over k, which is the familiar criterion for g to be diagonalizable over (We will extend this result in the next section.) Then p is separable in the usual Galois theory sense, its roots are in k, and g is diagonalizable over k,. 

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

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