nLab 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 $H\neq 0$ we have $Gr_k(H,\alpha_k)\neq 0$.

4. Any algebraic quotient of $G$ is an extension of subgroups of $\alpha_k$.

5. (If $p\neq 0)$, $\cap Im V^n_G =e$.

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