The Stacks project

Remark 114.15.6. Very reasonable algebraic spaces form a strictly larger collection than Zariski locally quasi-separated algebraic spaces. Consider an algebraic space of the form $X = [U/G]$ (see Spaces, Definition 64.14.4) where $G$ is a finite group acting without fixed points on a non-quasi-separated scheme $U$. Namely, in this case $U \times _ X U = U \times G$ and clearly both projections to $U$ are quasi-compact, hence $X$ is very reasonable. On the other hand, the diagonal $U \times _ X U \to U \times U$ is not quasi-compact, hence this algebraic space is not quasi-separated. Now, take $U$ the infinite affine space over a field $k$ of characteristic $\not= 2$ with zero doubled, see Schemes, Example 26.21.4. Let $0_1, 0_2$ be the two zeros of $U$. Let $G = \{ +1, -1\} $, and let $-1$ act by $-1$ on all coordinates, and by switching $0_1$ and $0_2$. Then $[U/G]$ is very reasonable but not Zariski locally quasi-separated (details omitted).


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