The Stacks project

Lemma 85.7.2. Let $\mathcal{C}_ n, f_\varphi , u_\varphi $ and $\mathcal{C}'_ n, f'_\varphi , u'_\varphi $ be as in Situation 85.3.3. Let $\mathcal{O}$ and $\mathcal{O}'$ be a sheaf of rings on $\mathcal{C}_{total}$ and $\mathcal{C}'_{total}$. Let $(h, h^\sharp )$ be a morphism between simplicial sites as in Remark 85.7.1. Then we obtain a morphism of ringed topoi

\[ h_{total} : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{total}, \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'_{total}), \mathcal{O}') \]

and commutative diagrams

\[ \xymatrix{ (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_ n), \mathcal{O}_ n) \ar[d]_{g_ n} \ar[r]_{h_ n} & (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'_ n), \mathcal{O}'_ n) \ar[d]^{g'_ n} \\ (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{total}), \mathcal{O}) \ar[r]^{h_{total}} & (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'_{total}), \mathcal{O}') } \]

of ringed topoi where $g_ n$ and $g'_ n$ are as in Lemma 85.6.1. Moreover, we have $(g'_ n)^* \circ h_{total, *} = h_{n, *} \circ g_ n^*$ as functor $\textit{Mod}(\mathcal{O}) \to \textit{Mod}(\mathcal{O}'_ n)$.

Proof. Follows from Lemma 85.5.2 and 85.6.1 by keeping track of the sheaves of rings. A small point is that in order to define $h_ n$ as a morphism of ringed topoi we set $h_ n^\sharp = g_ n^{-1}h^\sharp : g_ n^{-1}h_{total}^{-1}\mathcal{O}' \to g_ n^{-1}\mathcal{O}$ which makes sense because $g_ n^{-1}h_{total}^{-1}\mathcal{O}' = h_ n^{-1}(g'_ n)^{-1}\mathcal{O}' = h_ n^{-1}\mathcal{O}'_ n$ and $g_ n^{-1}\mathcal{O} = \mathcal{O}_ n$. Note that $g_ n^*\mathcal{F} = g_ n^{-1}\mathcal{F}$ for a sheaf of $\mathcal{O}$-modules $\mathcal{F}$ and similarly for $g'_ n$ and this helps explain why $(g'_ n)^* \circ h_{total, *} = h_{n, *} \circ g_ n^*$ follows from the corresponding statement of Lemma 85.5.2. $\square$

Comments (1)

Comment #8755 by ZL on


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Your email address will not be published.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 0DH0. Beware of the difference between the letter 'O' and the digit '0'.