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.

## 83.7 Morphisms of ringed simplicial sites

We continue the discussion of Section 83.5.

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

and commutative diagrams

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.
$\square$

## 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.

## Comments (0)