Example 63.14.1. Let $k$ be a field of characteristic $\not= 2$. Let $U = \mathbf{A}^1_ k$. Set

$j : R = \Delta \amalg \Gamma \longrightarrow U \times _ k U$

where $\Delta = \{ (x, x) \mid x \in \mathbf{A}^1_ k\}$ and $\Gamma = \{ (x, -x) \mid x \in \mathbf{A}^1_ k, x \not= 0\}$. It is clear that $s, t : R \to U$ are étale, and hence $j$ is an étale equivalence relation. The quotient $X = U/R$ is an algebraic space by Theorem 63.10.5. Since $R$ is quasi-compact we see that $X$ is quasi-separated. On the other hand, $X$ is not locally separated because the morphism $j$ is not an immersion.

Comments (2)

Comment #577 by Yogesh on

Since this is the first listed example of an algebraic space, can you add some detail (or links) on how it compares to the scheme $\mathbf{A}^1_k/\mu_2 = Spec (k[x^2])\simeq \mathbf{A}^1_k$, where $\mu_2=\{\pm 1 \}$ acts on $\mathbf{A}^1_k$ by $-1 \cdot x=-x$. I think there is a map from one to the other, but I'm not sure which way is goes.

Comment #593 by on

There is a map $X \to \text{Spec}(k[x^2])$ which is bijective but not an isomorphism. This is discussed in Remark 65.19.4 and Example 65.53.3.

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• 7 comment(s) on Section 63.14: Examples of algebraic spaces

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