The Stacks project

Lemma 84.7.2. Let $\mathcal{C}_ n, f_\varphi , u_\varphi $ and $\mathcal{C}'_ n, f'_\varphi , u'_\varphi $ be as in Situation 84.3.3. 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 84.7.1. 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 84.6.1. 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 84.5.2 and 84.6.1 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 84.5.2. $\square$

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