Lemma 109.47.2. 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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