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108.44 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

\[ G \times _ S G = U \times _ S U \cup V \times _ S U \cup U \times _ S V \cup V \times _ S V \]

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.

Lemma 108.44.1. There exists a flat group scheme of finite type over the affine line which is not separated.

Proof. See the discussion above. $\square$

Lemma 108.44.2. There exists a flat group scheme of finite type over the infinite dimensional affine space which is not quasi-separated.

Proof. The same construction as above can be carried out with the infinite dimensional affine space $S = \mathbf{A}^\infty _ k = \mathop{\mathrm{Spec}}k[x_1, x_2, \ldots ]$ as the base and the origin $0 \in S$ corresponding to the maximal ideal $(x_1, x_2, \ldots )$ as the closed point which is doubled in $G$. The resulting group scheme $G \rightarrow S$ is not quasi-separated as explained in Schemes, Example 26.21.4. $\square$


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