Lemma 17.28.7. Let $X$ be a topological space. Let $\mathcal{O}_1 \to \mathcal{O}_2$ be a homomorphism of sheaves of rings on $X$. Let $x \in X$. Then we have $\Omega _{\mathcal{O}_2/\mathcal{O}_1, x} = \Omega _{\mathcal{O}_{2, x}/\mathcal{O}_{1, x}}$.

**Proof.**
This is a special case of Lemma 17.28.6 for the inclusion map $\{ x\} \to X$. An alternative proof is to use Lemma 17.28.4, Sheaves, Lemma 6.17.2, and Algebra, Lemma 10.131.5.
$\square$

## 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Your email address will not be published.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 (5)

Comment #3610 by Herman Rohrbach on

Comment #3720 by Johan on

Comment #8564 by Elías Guisado on

Comment #8708 by Elías Guisado on

Comment #9143 by Stacks project on

There are also: