The Stacks project

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 $U \subset X$.


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