Lemma 110.48.1. There exists a flat group scheme of finite type over the affine line which is not separated.
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
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.
Proof. See the discussion above. \square
Lemma 110.48.2. 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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