The Stacks project

Lemma 17.29.5. Let $X$ be a topological space. Let $\mathcal{O}_1 \to \mathcal{O}_2$ be a homomorphism of presheaves of rings on $X$. Let $\mathcal{F}$ be a presheaf of $\mathcal{O}_2$-modules. Then $\mathcal{P}^ k_{\mathcal{O}_2^\# /\mathcal{O}_1^\# }(\mathcal{F}^\# )$ is the sheaf associated to the presheaf $U \mapsto P^ k_{\mathcal{O}_2(U)/\mathcal{O}_1(U)}(\mathcal{F}(U))$.

Proof. This can be proved in exactly the same way as is done for the sheaf of differentials in Lemma 17.28.4. Perhaps a more pleasing approach is to use the universal property of Lemma 17.29.3 directly to see the equality. We omit the details. $\square$

Comments (0)

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Your email address will not be published.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 0G3U. Beware of the difference between the letter 'O' and the digit '0'.