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Demazure, lectures on p-divisible groups, II.8, multiplicative affine groups

This entry is about a section of the text

• Michel Demazure, lectures on p-divisible groups, web

Remark

Let $G$ be a $k$-group-functor. Then the following conditions are equivalent:

1. $G$ is the Cartier dual of a constant group.

2. $G$ is an affine $k$-group and the $k$-ring $O(G)$ is generated by the morphisms $G\to {\mu }_k$ (these are called characters of $G$).

Definition

A $k$-group satisfying the conditions of the previous remark is called diagnalizable k-group.

Theorem

Let $G$ be a $k$-group. Then the following conditions are equivalent:

1. $G{\otimes }_k{k}_s$ is diagonalizable.

2. $G{\otimes }_{k}K$ is diagonalizable for a field $K\in M_k$.

3. $G$ is the Cartier dual of an étale $k$-group.

4. $\hat{D}(G)$ is an étale? $k$-formal group.

5. ${\mathrm{Gr}}_k(G,{\alpha }_k)=0$

6. (If $p\ne 0)$, $V_G$ is an epimorphism

7. (If $p\ne 0)$, $V_G$ is an isomorphism

Definition and Remark

1. A $k$-group satisfying the conditions of the previous theorem is called multiplicative k-group.

2. Multiplicative $k$-groups correspond by duality to étale formal $k$-groups.

3. The category ${\mathrm{ACm}}_k$ of multiplicative $k$-groups forma a subcategory of the category ${\mathrm{AC}}_k$ of affine commutative $k$-groups which is stable under forming subgroups, quatients, extensions (the set of these properties says that the subcategory is thick) and limits.

4. ${\mathrm{ACm}}_k$ is (contravariant) equivalent to the category of Galois modules: To $G$ corresponds the Galois module $\hat{D}(G{\otimes }_k{k}_s)(k_s)={\mathrm{Gr}}_{k_s}(G{\otimes }_k{k}_s,{\mu }_{k_s})$.

5. If $E$ is an étale $k$-group, then $D(E)$ is multiplicative and $\hat{D}(D(E))=E$. And we have $D(D(E))=E$. The duality is hence given by $E\to D(E)$ , $G\to D(G)$ without reference to formal groups.