The Stacks project

Example 21.38.3. Let $u : \mathcal{C}' \to \mathcal{C}$ be a functor. Endow $\mathcal{C}'$ and $\mathcal{C}$ with the chaotic topology as in Example 21.38.1. 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 21.37.3. Thus Lemma 21.37.5 applies to $g$ so we do not need to distinguish between $g_!$ on modules or abelian sheaves. In particular Remark 21.37.7 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 21.38.2 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.


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 08PH. Beware of the difference between the letter 'O' and the digit '0'.