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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


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