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 ith 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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