Lemma 50.8.1. In the situation above there is a canonical isomorphism

of complexes of $f^{-1}\mathcal{O}_ S$-modules.

Lemma 50.8.1. In the situation above there is a canonical isomorphism

\[ \text{Tot}(\Omega ^\bullet _{X/S} \boxtimes \Omega ^\bullet _{Y/S}) \longrightarrow \Omega ^\bullet _{X \times _ S Y/S} \]

of complexes of $f^{-1}\mathcal{O}_ S$-modules.

**Proof.**
We know that $ \Omega _{X \times _ S Y/S} = p^*\Omega _{X/S} \oplus q^*\Omega _{Y/S} $ by Morphisms, Lemma 29.32.11. Taking exterior powers we obtain

\[ \Omega ^ n_{X \times _ S Y/S} = \bigoplus \nolimits _{i + j = n} p^*\Omega ^ i_{X/S} \otimes _{\mathcal{O}_{X \times _ S Y}} q^*\Omega ^ j_{Y/S} = \bigoplus \nolimits _{i + j = n} \Omega ^ i_{X/S} \boxtimes \Omega ^ j_{Y/S} \]

by elementary properties of exterior powers. These identifications determine isomorphisms between the terms of the complexes on the left and the right of the arrow in the lemma. We omit the verification that these maps are compatible with differentials. $\square$

Your email address will not be published.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)