The Stacks project

110.48 A non-separated flat group scheme

Every group scheme over a field is separated, see Groupoids, Lemma 39.7.3. This is not true for group schemes over a base.

Let $k$ be a field. Let $S = \mathop{\mathrm{Spec}}(k[x]) = \mathbf{A}^1_ k$. Let $G$ be the affine line with $0$ doubled (see Schemes, Example 26.14.3) seen as a scheme over $S$. Thus a fibre of $G \to S$ is either a singleton or a set with two elements (one in $U$ and one in $V$). Thus we can endow these fibres with the structure of a group (by letting the element in $U$ be the zero of the group structure). More precisely, $G$ has two opens $U, V$ which map isomorphically to $S$ such that $U \cap V$ is mapped isomorphically to $S \setminus \{ 0\}$. Then

where each piece is isomorphic to $S$. Hence we can define a multiplication $m : G \times _ S G \to G$ as the unique $S$-morphism which maps the first and the last piece into $U$ and the two middle pieces into $V$. This matches the pointwise description given above. We omit the verification that this defines a group scheme structure.