The Stacks project

Remark 36.6.14. Let $g : X' \to X$ be a morphism of schemes. Let $f_1, \ldots , f_ c \in \Gamma (X, \mathcal{O}_ X)$. Set $f'_ i = g^\sharp (f_ i) \in \Gamma (X', \mathcal{O}_{X'})$. Denote $Z \subset X$, resp. $Z' \subset X'$ the closed subscheme cut out by $f_1, \ldots , f_ c$, resp. $f'_1, \ldots , f'_ c$. Then $Z' = Z \times _ X X'$. Denote $h : Z' \to Z$ the induced morphism of schemes. Let $\mathcal{F}$ be an $\mathcal{O}_ X$-module. Set $\mathcal{F}' = g^*\mathcal{F}$. In this setting, if $\mathcal{F}$ is quasi-coherent, then the diagram

\[ \xymatrix{ (i')^{-1}\mathcal{O}_{X'} \otimes _{h^{-1}i^{-1}\mathcal{O}_ X} h^{-1}\mathcal{H}^ c_ Z(\mathcal{F}) \ar[r] & \mathcal{H}_{Z'}^ c(\mathcal{F}') \\ h^*i^*\mathcal{F} \ar[r] \ar[u]_-{c_{f_1, \ldots , f_ c}} & (i')^*\mathcal{F}' \ar[u]^-{c_{f'_1, \ldots , f'_ c}} } \]

is commutative where the top horizonal arrow is the map of Cohomology, Remark 20.34.12 on cohomology sheaves in degree $c$. Namely, denote $\mathcal{F}^\bullet $, resp. $(\mathcal{F}')^\bullet $ the extended alternating Čech complex constructed in Remark 36.6.4 using $\mathcal{F}, f_1, \ldots , f_ c$, resp. $\mathcal{F}', f'_1, \ldots , f'_ c$. Note that $(\mathcal{F}')^\bullet = g^*\mathcal{F}^\bullet $. Then, without assuming $\mathcal{F}$ is quasi-coherent, the diagram

\[ \xymatrix{ i'_* L(g|_{Z'})^* R\mathcal{H}_ Z(\mathcal{F}) \ar[r] \ar@{=}[d] & i'_*R\mathcal{H}_{Z'}(Lg^*\mathcal{F}) \ar[d] \\ Lg^*i_*R\mathcal{H}_ Z(\mathcal{F}) & i'_*R\mathcal{H}_{Z'}(\mathcal{F}') \\ Lg^*(\mathcal{F}^\bullet ) \ar[u] \ar[r] & (\mathcal{F}')^\bullet \ar[u] } \]

is commutative where $g|_{Z'} : (Z', (i')^{-1}\mathcal{O}_{X'}) \to (Z, i^{-1}\mathcal{O}_ X)$ is the induced morphism of ringed spaces. Here the top horizontal arrow is given in Cohomology, Remark 20.34.12 as is the explanation for the equal sign. The arrows pointing up are from Remark 36.6.6. The lower horizonal arrow is the map $Lg^*\mathcal{F}^\bullet \to g^*\mathcal{F}^\bullet = (\mathcal{F}')^\bullet $ and the arrow pointing down is induced by $Lg^*\mathcal{F} \to g^*\mathcal{F} = \mathcal{F}'$. The diagram commutes because going around the diagram both ways we obtain two arrows $Lg^*\mathcal{F}^\bullet \to i'_*R\mathcal{H}_{Z'}(\mathcal{F}')$ whose composition with $i'_*R\mathcal{H}_{Z'}(\mathcal{F}') \to \mathcal{F}'$ is the canonical map $Lg^*\mathcal{F}^\bullet \to \mathcal{F}'$. Some details omitted. Now the commutativity of the first diagram follows by looking at this diagram on cohomology sheaves in degree $c$ and using that the construction of the map $i^*\mathcal{F} \to \mathop{\mathrm{Coker}}(\bigoplus \mathcal{F}_{1 \ldots \hat i \ldots c} \to \mathcal{F}_{1 \ldots c})$ used in Remark 36.6.10 is compatible with pullbacks.

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