The Stacks project

Example 52.27.7. The dimension bound in Proposition 52.27.6 is sharp. For example the Picard group of the punctured spectrum of $A = k[[x, y, z, w]]/(xy - zw)$ is nontrivial. Namely, the ideal $I = (x, z)$ cuts out an effective Cartier divisor $D$ on the punctured spectrum $U$ of $A$ as it is easy to see that $I_ x, I_ y, I_ z, I_ w$ are invertible ideals in $A_ x, A_ y, A_ z, A_ w$. But on the other hand, $A/I$ has depth $\geq 1$ (in fact $2$), hence $I$ has depth $\geq 2$ (in fact $3$), hence $I = \Gamma (U, \mathcal{O}_ U(-D))$. Thus if $\mathcal{O}_ U(-D)$ were trivial, then we'd have $I \cong \Gamma (U, \mathcal{O}_ U) = A$ which isn't true as $I$ isn't generated by $1$ element.

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