The Stacks project

Exercise 110.40.8. Let $R$ be a ring.

  1. Show that if $R$ is a Noetherian normal domain, then $\mathop{\mathrm{Pic}}\nolimits (R) = \mathop{\mathrm{Pic}}\nolimits (R[t])$. [Hint: There is a map $R[t] \to R$, $t \mapsto 0$ which is a left inverse to the map $R \to R[t]$. Hence it suffices to show that any invertible $R[t]$-module $M$ such that $M/tM \cong R$ is free of rank $1$. Let $K$ be the fraction field of $R$. Pick a trivialization $K[t] \to M \otimes _{R[t]} K[t]$ which is possible by Exercise 110.40.5 (1). Adjust it so it agrees with the trivialization of $M/tM$ above. Show that it is in fact a trivialization of $M$ over $R[t]$ (this is where normality comes in).]

  2. Let $k$ be a field. Show that $\mathop{\mathrm{Pic}}\nolimits (k[x^2, x^3, t]) \not= \mathop{\mathrm{Pic}}\nolimits (k[x^2, x^3])$.


Comments (0)

There are also:

  • 2 comment(s) on Section 110.40: Invertible sheaves

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Your email address will not be published.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 02AN. Beware of the difference between the letter 'O' and the digit '0'.