Example 65.14.1. 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 65.10.5. 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 67.19.4 and Example 67.53.3.

There are also:

• 9 comment(s) on Section 65.14: Examples of algebraic spaces

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 02Z1. Beware of the difference between the letter 'O' and the digit '0'.