The Stacks project

Remark 36.6.10. With $X$, $f_1, \ldots , f_ c \in \Gamma (X, \mathcal{O}_ X)$, and $\mathcal{F}$ as in Remark 36.6.4. There is a canonical $\mathcal{O}_ X|_ Z$-linear map

functorial in $\mathcal{F}$. Namely, denoting $\mathcal{F}^\bullet$ the extended alternating Čech complex (36.6.4.1) we have the canonical map $\mathcal{F}^\bullet \to i_*R\mathcal{H}_ Z(\mathcal{F})$ of Remark 36.6.6. This determines a canonical map

on cohomology sheaves in degree $c$. Given a local section $s$ of $\mathcal{F}$ we can consider the local section

of $\mathcal{F}_{1 \ldots c}$. The class of this section in the cokernel displayed above depends only on $s$ modulo the image of $(f_1, \ldots , f_ c) : \mathcal{F}^{\oplus c} \to \mathcal{F}$. Since $i_*i^*\mathcal{F}$ is equal to the cokernel of $(f_1, \ldots , f_ c) : \mathcal{F}^{\oplus c} \to \mathcal{F}$ we see that we get an $\mathcal{O}_ X$-module map $i_*i^*\mathcal{F} \to i_*\mathcal{H}_ Z^ c(\mathcal{F})$. As $i_*$ is fully faithful we get the map $c_{f_1, \ldots , f_ c}$.