The Stacks project

83.7 Morphisms of ringed simplicial sites

We continue the discussion of Section 83.5.

Remark 83.7.1. Let $\mathcal{C}_ n, f_\varphi , u_\varphi $ and $\mathcal{C}'_ n, f'_\varphi , u'_\varphi $ be as in Situation 83.3.3. Let $\mathcal{O}$ and $\mathcal{O}'$ be a sheaf of rings on $\mathcal{C}_{total}$ and $\mathcal{C}'_{total}$. We will say that $(h, h^\sharp )$ is a morphism between ringed simplicial sites if $h$ is a morphism between simplicial sites as in Remark 83.5.1 and $h^\sharp : h_{total}^{-1}\mathcal{O}' \to \mathcal{O}$ or equivalently $h^\sharp : \mathcal{O}' \to h_{total, *}\mathcal{O}$ is a homomorphism of sheaves of rings.

Lemma 83.7.2. Let $\mathcal{C}_ n, f_\varphi , u_\varphi $ and $\mathcal{C}'_ n, f'_\varphi , u'_\varphi $ be as in Situation 83.3.3. 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 83.7.1. 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 83.6.1. 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 83.5.2 and 83.6.1 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 83.5.2. $\square$

Lemma 83.7.3. With notation and hypotheses as in Lemma 83.7.2. For $K \in D(\mathcal{O})$ we have $(g'_ n)^*Rh_{total, *}K = Rh_{n, *}g_ n^*K$.

Proof. Recall that $g_ n^* = g_ n^{-1}$ because $g_ n^{-1}\mathcal{O} = \mathcal{O}_ n$ by the construction in Lemma 83.6.1. In particular $g_ n^*$ is exact and $Lg_ n^*$ is given by applying $g_ n^*$ to any representative complex of modules. Similarly for $g'_ n$. There is a canonical base change map $(g'_ n)^*Rh_{total, *}K \to Rh_{n, *}g_ n^*K$, see Cohomology on Sites, Remark 21.19.3. By Cohomology on Sites, Lemma 21.20.7 the image of this in $D(\mathcal{C}'_ n)$ is the map $(g'_ n)^{-1}Rh_{total, *}K_{ab} \to Rh_{n, *}g_ n^{-1}K_{ab}$ where $K_{ab}$ is the image of $K$ in $D(\mathcal{C}_{total})$. This we proved to be an isomorphism in Lemma 83.5.3 and the result follows. $\square$


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