17.30 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 17.28. 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 17.30.1. 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
17.30.1.1
\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 17.28.4. 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 17.30.2. 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 17.28.6.
$\square$
Lemma 17.30.3. 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 17.28, 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 (17.30.1.1).
Definition 17.30.4. 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 17.30.2 for the first equality) and then we can use adjunction.
Lemma 17.30.5. 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 17.30.3 and the definition.
$\square$
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