The Stacks project

Remark 36.6.4. Let $X$ be a scheme. Let $f_1, \ldots , f_ c \in \Gamma (X, \mathcal{O}_ X)$. Denote $Z \subset X$ the closed subscheme cut out by $f_1, \ldots , f_ c$. For $0 \leq p < c$ and $1 \leq i_0 < \ldots < i_ p \leq c$ we denote $U_{i_0 \ldots i_ p} \subset X$ the open subscheme where $f_{i_0} \ldots f_{i_ p}$ is invertible. For any $\mathcal{O}_ X$-module $\mathcal{F}$ we set

\[ \mathcal{F}_{i_0 \ldots i_ p} = (U_{i_0 \ldots i_ p} \to X)_*(\mathcal{F}|_{U_{i_0 \ldots i_ p}}) \]

In this situation the extended alternating Čech complex is the complex of $\mathcal{O}_ X$-modules
\begin{equation} \label{perfect-equation-extended-alternating} 0 \to \mathcal{F} \to \bigoplus \nolimits _{i_0} \mathcal{F}_{i_0} \to \ldots \to \bigoplus \nolimits _{i_0 < \ldots < i_ p} \mathcal{F}_{i_0 \ldots i_ p} \to \ldots \to \mathcal{F}_{1 \ldots c} \to 0 \end{equation}

where $\mathcal{F}$ is put in degree $0$. The maps are constructed as follows. Given $1 \leq i_0 < \ldots < i_{p + 1} \leq c$ and $0 \leq j \leq p + 1$ we have the canonical map

\[ \mathcal{F}_{i_0 \ldots \hat i_ j \ldots i_{p + 1}} \to \mathcal{F}_{i_0 \ldots i_ p} \]

coming from the inclusion $U_{i_0 \ldots i_ p} \subset U_{i_0 \ldots \hat i_ j \ldots i_{p + 1}}$. The differentials in the extended alternating complex use these canonical maps with sign $(-1)^ j$.

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