Lemma 50.3.1. Let $X \to S$ be a morphism of affine schemes given by the ring map $R \to A$. Then $R\Gamma (X, \Omega ^\bullet _{X/S}) = \Omega ^\bullet _{A/R}$ in $D(R)$ and $H^ i_{dR}(X/S) = H^ i(\Omega ^\bullet _{A/R})$.
50.3 de Rham cohomology
Let $p : X \to S$ be a morphism of schemes. We define the de Rham cohomology of $X$ over $S$ to be the cohomology groups
\[ H^ i_{dR}(X/S) = H^ i(R\Gamma (X, \Omega ^\bullet _{X/S})) \]
Since $\Omega ^\bullet _{X/S}$ is a complex of $p^{-1}\mathcal{O}_ S$-modules, these cohomology groups are naturally modules over $H^0(S, \mathcal{O}_ S)$.
Given a commutative diagram
\[ \xymatrix{ X' \ar[r]_ f \ar[d] & X \ar[d] \\ S' \ar[r] & S } \]
of schemes, using the canonical maps of Section 50.2 we obtain pullback maps
\[ f^* : R\Gamma (X, \Omega ^\bullet _{X/S}) \longrightarrow R\Gamma (X', \Omega ^\bullet _{X'/S'}) \]
and
\[ f^* : H^ i_{dR}(X/S) \longrightarrow H^ i_{dR}(X'/S') \]
These pullbacks satisfy an obvious composition law. In particular, if we work over a fixed base scheme $S$, then de Rham cohomology is a contravariant functor on the category of schemes over $S$.
Proof. This follows from Cohomology of Schemes, Lemma 30.2.2 and Leray's acyclicity lemma (Derived Categories, Lemma 13.16.7). $\square$
Lemma 50.3.2. Let $p : X \to S$ be a morphism of schemes. If $p$ is quasi-compact and quasi-separated, then $Rp_*\Omega ^\bullet _{X/S}$ is an object of $D_\mathit{QCoh}(\mathcal{O}_ S)$.
Proof. There is a spectral sequence with first page $E_1^{a, b} = R^ bp_*\Omega ^ a_{X/S}$ converging to the cohomology of $Rp_*\Omega ^\bullet _{X/S}$ (see Derived Categories, Lemma 13.21.3). Hence by Homology, Lemma 12.25.3 it suffices to show that $R^ bp_*\Omega ^ a_{X/S}$ is quasi-coherent. This follows from Cohomology of Schemes, Lemma 30.4.5. $\square$
Lemma 50.3.3. Let $p : X \to S$ be a proper morphism of schemes with $S$ locally Noetherian. Then $Rp_*\Omega ^\bullet _{X/S}$ is an object of $D_{\textit{Coh}}(\mathcal{O}_ S)$.
Proof. In this case by Morphisms, Lemma 29.32.12 the modules $\Omega ^ i_{X/S}$ are coherent. Hence we can use exactly the same argument as in the proof of Lemma 50.3.2 using Cohomology of Schemes, Proposition 30.19.1. $\square$
Lemma 50.3.4. Let $A$ be a Noetherian ring. Let $X$ be a proper scheme over $S = \mathop{\mathrm{Spec}}(A)$. Then $H^ i_{dR}(X/S)$ is a finite $A$-module for all $i$.
Proof. This is a special case of Lemma 50.3.3. $\square$
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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