The Stacks project

Example 70.6.10. This is a continuation of Morphisms of Spaces, Example 65.53.3. Consider the algebraic space $X = \mathbf{A}^1_ k/\{ t \sim -t \mid t \not= 0\} $. This is a smooth algebraic space over the field $k$. There is a universal homeomorphism

\[ X \longrightarrow \mathbf{A}^1_ k = \mathop{\mathrm{Spec}}(k[t]) \]

which is an isomorphism over $\mathbf{A}^1_ k \setminus \{ 0\} $. We conclude that $X$ is Noetherian and integral. Since $\dim (X) = 1$, we see that the prime divisors of $X$ are the closed points of $X$. Consider the unique closed point $x \in |X|$ lying over $0 \in \mathbf{A}^1_ k$. Since $X \setminus \{ x\} $ maps isomorphically to $\mathbf{A}^1 \setminus \{ 0\} $ we see that the classes in $\text{Cl}(X)$ of closed points different from $x$ are zero. However, the divisor of $t$ on $X$ is $2[x]$. We conclude that $\text{Cl}(X) = \mathbf{Z}/2\mathbf{Z}$.


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