The Stacks project

Lemma 50.8.2. Assume $X$ and $Y$ are smooth, quasi-compact, with affine diagonal over $S = \mathop{\mathrm{Spec}}(A)$. Then the map

\[ R\Gamma (X, \Omega ^\bullet _{X/S}) \otimes _ A^\mathbf {L} R\Gamma (Y, \Omega ^\bullet _{Y/S}) \longrightarrow R\Gamma (X \times _ S Y, \Omega ^\bullet _{X \times _ S Y/S}) \]

is an isomorphism in $D(A)$.

Proof. By Morphisms, Lemma 29.34.12 the sheaves $\Omega ^ n_{X/S}$ and $\Omega ^ m_{Y/S}$ are finite locally free $\mathcal{O}_ X$ and $\mathcal{O}_ Y$-modules. On the other hand, $X$ and $Y$ are flat over $S$ (Morphisms, Lemma 29.34.9) and hence we find that $\Omega ^ n_{X/S}$ and $\Omega ^ m_{Y/S}$ are flat over $S$. Also, observe that $\Omega ^\bullet _{X/S}$ is a locally bounded. Thus the result by Lemma 50.8.1 and Derived Categories of Schemes, Lemma 36.24.1. $\square$

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