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

\[ c_{f_1, \ldots , f_ c} : i^*\mathcal{F} \longrightarrow \mathcal{H}^ c_ Z(\mathcal{F}) \]

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

\[ \mathop{\mathrm{Coker}}\left(\bigoplus \mathcal{F}_{1 \ldots \hat i \ldots c} \to \mathcal{F}_{1 \ldots c}\right) \longrightarrow i_*\mathcal{H}^ c_ Z(\mathcal{F}) \]

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

\[ \frac{s}{f_1 \ldots f_ c} \]

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 $c_{f_1, \ldots , f_ c}$.

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  • 3 comment(s) on Section 36.6: Cohomology with support in a closed subset

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