The Stacks project

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. $\square$

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