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unipotent group scheme

Idea

An element $r$ of a ring with multiplicative unit is called unipotent element if $r-1$ is nilpotent.

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Theorem and Definition

Let $G$ be an affine k-group. Then the following conditions are equivalent.

1. The completion of the Cartier dual $\hat{D}(G)$ of $G$ is a connected formal group.

2. Any multiplicative subgroup of $G$ is zero.

3. For any subgroup $H$ of $G$ with $H\ne 0$ we have ${\mathrm{Gr}}_k(H,{\alpha }_k)\ne 0$.

4. Any algebraic quotient of $G$ is an extension of subgroups of ${\alpha }_k$.

5. (If $p\ne 0)$, $\cap \mathrm{Im}{V}_G^n=e$.

An affine group scheme satisfying these conditions is called unipotent group scheme.