The Stacks project

Lemma 17.28.8. Let $X$ be a topological space. Let

\[ \xymatrix{ \mathcal{O}_2 \ar[r]_\varphi & \mathcal{O}_2' \\ \mathcal{O}_1 \ar[r] \ar[u] & \mathcal{O}'_1 \ar[u] } \]

be a commutative diagram of sheaves of rings on $X$. The map $\mathcal{O}_2 \to \mathcal{O}'_2$ composed with the map $\text{d} : \mathcal{O}'_2 \to \Omega _{\mathcal{O}'_2/\mathcal{O}'_1}$ is a $\mathcal{O}_1$-derivation. Hence we obtain a canonical map of $\mathcal{O}_2$-modules $\Omega _{\mathcal{O}_2/\mathcal{O}_1} \to \Omega _{\mathcal{O}'_2/\mathcal{O}'_1}$. It is uniquely characterized by the property that $\text{d}(f) \mapsto \text{d}(\varphi (f))$ for any local section $f$ of $\mathcal{O}_2$. In this way $\Omega _{-/-}$ becomes a functor on the category of arrows of sheaves of rings.

Proof. This lemma proves itself. $\square$

Comments (0)

There are also:

  • 2 comment(s) on Section 17.28: Modules of differentials

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Your email address will not be published.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 08RS. Beware of the difference between the letter 'O' and the digit '0'.