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

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

of schemes, using the canonical maps of Section 50.2 we obtain pullback maps

and

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

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 20.46.7) 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. $\square$

## Post a comment

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 (3)

Comment #6522 by Hung Chiang on

Comment #6523 by Hung Chiang on

Comment #6579 by Johan on