Lemma 49.12.3. Let $f : Y \to X$ be a morphism of Noetherian schemes. If $f$ satisfies the equivalent conditions of Lemma 49.10.1 then the different $\mathfrak {D}_ f$ of $f$ is the Kähler different of $f$.

**Proof.**
By Lemmas 49.9.3 and 49.10.4 the different of $f$ affine locally is the same as the Noether different. Then the lemma follows from the computation of the Noether different and the Kähler different on standard affine pieces done in Lemmas 49.7.4 and 49.12.2.
$\square$

## 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Your email address will not be published.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 (0)