The Stacks project

Lemma 50.3.5. Let $f : X \to S$ be a proper smooth morphism of schemes. Then $Rf_*\Omega ^ p_{X/S}$, $p \geq 0$ and $Rf_*\Omega ^\bullet _{X/S}$ are perfect objects of $D(\mathcal{O}_ S)$ whose formation commutes with arbitrary change of base.

Proof. Since $f$ is smooth the modules $\Omega ^ p_{X/S}$ are finite locally free $\mathcal{O}_ X$-modules, see Morphisms, Lemma 29.34.12. Their formation commutes with arbitrary change of base by Lemma 50.2.1. Hence $Rf_*\Omega ^ p_{X/S}$ is a perfect object of $D(\mathcal{O}_ S)$ whose formation commutes with abitrary base change, see Derived Categories of Schemes, Lemma 36.30.4. This proves the first assertion of the lemma.

To prove that $Rf_*\Omega ^\bullet _{X/S}$ is perfect on $S$ we may work locally on $S$. Thus we may assume $S$ is quasi-compact. This means we may assume that $\Omega ^ n_{X/S}$ is zero for $n$ large enough. For every $p \geq 0$ we claim that $Rf_*\sigma _{\geq p}\Omega ^\bullet _{X/S}$ is a perfect object of $D(\mathcal{O}_ S)$ whose formation commutes with arbitrary change of base. By the above we see that this is true for $p \gg 0$. Suppose the claim holds for $p$ and consider the distinguished triangle

\[ \sigma _{\geq p}\Omega ^\bullet _{X/S} \to \sigma _{\geq p - 1}\Omega ^\bullet _{X/S} \to \Omega ^{p - 1}_{X/S}[-(p - 1)] \to (\sigma _{\geq p}\Omega ^\bullet _{X/S})[1] \]

in $D(f^{-1}\mathcal{O}_ S)$. Applying the exact functor $Rf_*$ we obtain a distinguished triangle in $D(\mathcal{O}_ S)$. Since we have the 2-out-of-3 property for being perfect (Cohomology, Lemma 20.49.7) we conclude $Rf_*\sigma _{\geq p - 1}\Omega ^\bullet _{X/S}$ is a perfect object of $D(\mathcal{O}_ S)$. Similarly for the commutation with arbitrary base change. $\square$

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