The Stacks project

Example 21.39.3. Let $u : \mathcal{C}' \to \mathcal{C}$ be a functor. Endow $\mathcal{C}'$ and $\mathcal{C}$ with the chaotic topology as in Example 21.39.1. The functors $u$, $\mathcal{C}' \to *$, and $\mathcal{C} \to *$ where $*$ is the category with a single object and a single morphism are cocontinuous and continuous. Let $g : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}') \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$, $\pi ' : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}') \to \mathop{\mathit{Sh}}\nolimits (*)$, and $\pi : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (*)$, be the corresponding morphisms of topoi. Let $B$ be a ring. We endow $*$ with the sheaf of rings $B$ and $\mathcal{C}'$, $\mathcal{C}$ with the constant sheaf $\underline{B}$. In this way

\[ \xymatrix{ (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'), \underline{B}) \ar[rd]_{\pi '} \ar[rr]_ g & & (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \underline{B}) \ar[ld]^\pi \\ & (\mathop{\mathit{Sh}}\nolimits (*), B) } \]

is an example of Situation 21.38.3. Thus Lemma 21.38.5 applies to $g$ so we do not need to distinguish between $g_!$ on modules or abelian sheaves. In particular Remark 21.38.7 produces canonical maps

\[ H_ n(\mathcal{C}', \mathcal{F}') \longrightarrow H_ n(\mathcal{C}, \mathcal{F}) \]

whenever we have $\mathcal{F}$ in $\textit{Ab}(\mathcal{C})$, $\mathcal{F}'$ in $\textit{Ab}(\mathcal{C}')$, and a map $t : \mathcal{F}' \to g^{-1}\mathcal{F}$. In terms of the computation of homology given in Example 21.39.2 we see that these maps come from a map of complexes

\[ K_\bullet (\mathcal{F}') \longrightarrow K_\bullet (\mathcal{F}) \]

given by the rule

\[ (U'_ n \to \ldots \to U'_0, s') \longmapsto (u(U'_ n) \to \ldots \to u(U'_0), t(s')) \]

with obvious notation.

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