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.

## 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Your email address will not be published.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.

## Comments (0)