The Stacks project

Example 70.6.11. Let $k$ be a field. Let

\[ U = \mathop{\mathrm{Spec}}(k[x, y]/(xy)) \]

be the union of the coordinate axes in $\mathbf{A}^2_ k$. Denote $\Delta : U \to U \times _ k U$ the diagonal and $\Delta ' : U \to U \times _ k U$ the map $u \mapsto (u, \sigma (u))$ where $\sigma : U \to U$, $(x, y) \mapsto (y, x)$ is the automorphism flipping the coordinate axes. Set

\[ R = \Delta (U) \amalg \Delta '(U \setminus \{ 0_ U\} ) \]

where $0_ U \in U$ is the origin. It is easy to see that $R$ is an ├ętale equivalence relation on $U$. The quotient $X = U/R$ is an algebraic space. The morphism $U \to \mathbf{A}^1_ k$, $(x, y) \mapsto x + y$ is $R$-invariant and hence defines a morphism

\[ X \longrightarrow \mathbf{A}^1_ k \]

This morphism is a universal homeomorphism and an isomorphism over $\mathbf{A}^1_ k \setminus \{ 0\} $. It follows that $X$ is integral and Noetherian. Exactly as in Example 70.6.10 the reader shows that $\text{Cl}(X) = \mathbf{Z}/2\mathbf{Z}$ with generator corresponding to the unique closed point $x \in |X|$ mapping to $0 \in \mathbf{A}^1_ k$. However, in this case the henselian local ring of $X$ at $x$ isn't a domain, as it is the henselization of $\mathcal{O}_{U, 0_ U}$.

Comments (0)

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Your email address will not be published.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 0ENU. Beware of the difference between the letter 'O' and the digit '0'.