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 $\mathcal{O}_{U, 0_ U}$.


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