The Stacks project

17.29 The de Rham complex

The section is the analogue of Algebra, Section 10.132 for morphisms of ringed spaces. We urge the reader to read that section first.

Let $X$ be a topological space. Let $\mathcal{A} \to \mathcal{B}$ be a homomorphism of sheaves of rings. Denote $\text{d} : \mathcal{B} \to \Omega _{\mathcal{B}/\mathcal{A}}$ the module of differentials with its universal $\mathcal{A}$-derivation constructed in Section 17.27. Let

\[ \Omega _{\mathcal{B}/\mathcal{A}}^ i = \wedge ^ i_\mathcal {B}(\Omega _{\mathcal{B}/\mathcal{A}}) \]

for $i \geq 0$ be the $i$th exterior power as in Section 17.21.

Definition 17.29.1. In the situation above, the de Rham complex of $\mathcal{B}$ over $\mathcal{A}$ is the unique complex

\[ \Omega _{\mathcal{B}/\mathcal{A}}^0 \to \Omega _{\mathcal{B}/\mathcal{A}}^1 \to \Omega _{\mathcal{B}/\mathcal{A}}^2 \to \ldots \]

of sheaves of $\mathcal{A}$-modules whose differential in degree $0$ is given by $\text{d} : \mathcal{B} \to \Omega _{\mathcal{B}/\mathcal{A}}$ and whose differentials in higher degrees have the following property

17.29.1.1
\begin{equation} \label{modules-equation-rule} \text{d}\left(b_0\text{d}b_1 \wedge \ldots \wedge \text{d}b_ p\right) = \text{d}b_0 \wedge \text{d}b_1 \wedge \ldots \wedge \text{d}b_ p \end{equation}

where $b_0, \ldots , b_ p \in \mathcal{B}(U)$ are sections over a common open $U \subset X$.

We could construct this complex by repeating the cumbersome arguments given in Algebra, Section 10.132. Instead we recall that $\Omega _{\mathcal{B}/\mathcal{A}}$ is the sheafification of the presheaf $U \mapsto \Omega _{\mathcal{B}(U)/\mathcal{A}(U)}$, see Lemma 17.27.4. Thus $\Omega _{\mathcal{B}/\mathcal{A}}^ i$ is the sheafification of the presheaf $U \mapsto \Omega ^ i_{\mathcal{B}(U)/\mathcal{A}(U)}$, see Lemma 17.21.1. Therefore we can define the de Rham complex as the sheafification of the rule

\[ U \longmapsto \Omega ^\bullet _{\mathcal{B}(U)/\mathcal{A}(U)} \]

Lemma 17.29.2. Let $f : Y \to X$ be a continuous map of topological spaces. Let $\mathcal{A} \to \mathcal{B}$ be a homomorphism of sheaves of rings on $X$. Then there is a canonical identification $f^{-1}\Omega ^\bullet _{\mathcal{B}/\mathcal{A}} = \Omega ^\bullet _{f^{-1}\mathcal{B}/f^{-1}\mathcal{A}}$ of de Rham complexes.

Proof. Omitted. Hint: compare with Lemma 17.27.6. $\square$

Lemma 17.29.3. Let $X$ be a topological space. Let $\mathcal{A} \to \mathcal{B}$ be a homomorphism of sheaves of rings on $X$. The differentials $\text{d} : \Omega ^ i_{\mathcal{B}/\mathcal{A}} \to \Omega ^{i + 1}_{\mathcal{B}/\mathcal{A}}$ are differential operators of order $1$.

Proof. Via our construction of the de Rham complex above as the sheafification of the rule $U \mapsto \Omega ^\bullet _{\mathcal{B}(U)/\mathcal{A}(U)}$ this follows from Algebra, Lemma 10.133.8. $\square$

Let $X$ be a topological space. Let

\[ \xymatrix{ \mathcal{B} \ar[r] & \mathcal{B}' \\ \mathcal{A} \ar[r] \ar[u] & \mathcal{A}' \ar[u] } \]

be a commutative diagram of sheaves of rings on $X$. There is a natural map of de Rham complexes

\[ \Omega ^\bullet _{\mathcal{B}/\mathcal{A}} \longrightarrow \Omega ^\bullet _{\mathcal{B}'/\mathcal{A}'} \]

Namely, in degree $0$ this is the map $\mathcal{B} \to \mathcal{B}'$, in degree $1$ this is the map $\Omega _{\mathcal{B}/\mathcal{A}} \to \Omega _{\mathcal{B}'/\mathcal{A}'}$ constructed in Section 17.27, and for $p \geq 2$ it is the induced map $\Omega ^ p_{\mathcal{B}/\mathcal{A}} = \wedge ^ p_\mathcal {B}(\Omega _{\mathcal{B}/\mathcal{A}}) \to \wedge ^ p_{\mathcal{B}'}(\Omega _{\mathcal{B}'/\mathcal{A}'}) = \Omega ^ p_{\mathcal{B}'/\mathcal{A}'}$. The compatibility with differentials follows from the characterization of the differentials by the formula (17.29.1.1).

Definition 17.29.4. Let $f : (X, \mathcal{O}_ X) \to (Y, \mathcal{O}_ Y)$ be a morphism of ringed spaces. The de Rham complex of $f$ or of $X$ over $Y$ is the complex

\[ \Omega ^\bullet _{X/Y} = \Omega ^\bullet _{\mathcal{O}_ X/f^{-1}\mathcal{O}_ Y} \]

Consider a commutative diagram of ringed spaces

\[ \xymatrix{ X' \ar[d]_{h'} \ar[r]_ f & X \ar[d]^ h \\ S' \ar[r]^ g & S } \]

Then we obtain a canonical map

\[ \Omega ^\bullet _{X/S} \to f_*\Omega ^\bullet _{X'/S'} \]

of de Rham complexes. Namely, the commutative diagram of sheaves of rings

\[ \xymatrix{ f^{-1}\mathcal{O}_ X \ar[r] & \mathcal{O}_{X'} \\ f^{-1}h^{-1}\mathcal{O}_ S \ar[u] \ar[r] & (h')^{-1}\mathcal{O}_{S'} \ar[u] } \]

on $X'$ produces a map of complexes (see above)

\[ f^{-1}\Omega ^\bullet _{X/S} = \Omega ^\bullet _{f^{-1}\mathcal{O}_ X/f^{-1}h^{-1}\mathcal{O}_ S} \longrightarrow \Omega ^\bullet _{\mathcal{O}_{X'}/(h')^{-1}\mathcal{O}_{S'}} = \Omega ^\bullet _{X'/S'} \]

(using Lemma 17.29.2 for the first equality) and then we can use adjunction.

Lemma 17.29.5. Let $f : X \to Y$ be a morphism of ringed spaces. The differentials $\text{d} : \Omega ^ i_{X/Y} \to \Omega ^{i + 1}_{X/Y}$ are differential operators of order $1$ on $X/Y$.

Proof. Immediate from Lemma 17.29.3 and the definition. $\square$


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