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Let be a -group-functor. Then the following conditions are equivalent:
A -group satisfying the conditions of the previous remark is called diagnalizable k-group.
Let be a -group. Then the following conditions are equivalent:
A -group satisfying the conditions of the previous theorem is called multiplicative k-group.
Multiplicative -groups correspond by duality to étale formal -groups.
The category of multiplicative -groups forma a subcategory of the category of affine commutative -groups which is stable under forming subgroups, quatients, extensions (the set of these properties says that the subcategory is thick) and limits.
is (contravariant) equivalent to the category of Galois modules: To corresponds the Galois module .
If is an étale -group, then is multiplicative and . And we have . The duality is hence given by , without reference to formal groups.