The Stacks project

84.6 Ringed simplicial sites

Let us endow our simplicial topos with a sheaf of rings.

Lemma 84.6.1. In Situation 84.3.3. Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}_{total}$. There is a canonical morphism of ringed topoi $g_ n : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_ n), \mathcal{O}_ n) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{total}), \mathcal{O})$ agreeing with the morphism $g_ n$ of Lemma 84.3.5 on underlying topoi. The functor $g_ n^* : \textit{Mod}(\mathcal{O}) \to \textit{Mod}(\mathcal{O}_ n)$ has a left adjoint $g_{n!}$. For $\mathcal{G}$ in $\textit{Mod}(\mathcal{O}_ n)$-modules the restriction of $g_{n!}\mathcal{G}$ to $\mathcal{C}_ m$ is

\[ \bigoplus \nolimits _{\varphi : [n] \to [m]} f_\varphi ^*\mathcal{G} \]

where $f_\varphi : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_ m), \mathcal{O}_ m) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_ n), \mathcal{O}_ n)$ is the morphism of ringed topoi agreeing with the previously defined $f_\varphi $ on topoi and using the map $\mathcal{O}(\varphi ) : f_\varphi ^{-1}\mathcal{O}_ n \to \mathcal{O}_ m$ on sheaves of rings.

Proof. By Lemma 84.3.5 we have $g_ n^{-1}\mathcal{O} = \mathcal{O}_ n$ and hence we obtain our morphism of ringed topoi. By Modules on Sites, Lemma 18.41.1 we obtain the adjoint $g_{n!}$. To prove the formula for $g_{n!}$ we first define a sheaf of $\mathcal{O}$-modules $\mathcal{H}$ on $\mathcal{C}_{total}$ with degree $m$ component the $\mathcal{O}_ m$-module

\[ \mathcal{H}_ m = \bigoplus \nolimits _{\varphi : [n] \to [m]} f_\varphi ^*\mathcal{G} \]

Given a map $\psi : [m] \to [m']$ the map $\mathcal{H}(\psi ) : f_\psi ^{-1}\mathcal{H}_ m \to \mathcal{H}_{m'}$ is given on components by

\[ f_\psi ^{-1} f_\varphi ^*\mathcal{G} \to f_\psi ^* f_\varphi ^*\mathcal{G} \to f_{\psi \circ \varphi }^*\mathcal{G} \]

Since this map $f_\psi ^{-1}\mathcal{H}_ m \to \mathcal{H}_{m'}$ is $\mathcal{O}(\psi ) : f_\psi ^{-1}\mathcal{O}_ m \to \mathcal{O}_{m'}$-semi-linear, this indeed does define an $\mathcal{O}$-module (use Lemma 84.3.4). Then one proves directly that

\[ \mathop{\mathrm{Mor}}\nolimits _{\mathcal{O}_ n}(\mathcal{G}, \mathcal{F}_ n) = \mathop{\mathrm{Mor}}\nolimits _{\mathcal{O}}(\mathcal{H}, \mathcal{F}) \]

proceeding as in the proof of Lemma 84.3.5. Thus $\mathcal{H} = g_{n!}\mathcal{G}$ as desired. $\square$

Lemma 84.6.2. In Situation 84.3.3. Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}_{total}$. If $\mathcal{I}$ is injective in $\textit{Mod}(\mathcal{O})$, then $\mathcal{I}_ n$ is a totally acyclic sheaf on $\mathcal{C}_ n$.

Proof. This follows from Cohomology on Sites, Lemma 21.37.4 applied to the inclusion functor $\mathcal{C}_ n \to \mathcal{C}_{total}$ and its properties proven in Lemma 84.3.5. $\square$

Lemma 84.6.3. With assumptions as in Lemma 84.6.1 the functor $g_{n!} : \textit{Mod}(\mathcal{O}_ n) \to \textit{Mod}(\mathcal{O})$ is exact if the maps $f_\varphi ^{-1}\mathcal{O}_ n \to \mathcal{O}_ m$ are flat for all $\varphi : [n] \to [m]$.

Proof. Recall that $g_{n!}\mathcal{G}$ is the $\mathcal{O}$-module whose degree $m$ part is the $\mathcal{O}_ m$-module

\[ \bigoplus \nolimits _{\varphi : [n] \to [m]} f_\varphi ^*\mathcal{G} \]

Here the morphism of ringed topoi $f_\varphi : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_ m), \mathcal{O}_ m) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_ n), \mathcal{O}_ n)$ uses the map $f_\varphi ^{-1}\mathcal{O}_ n \to \mathcal{O}_ m$ of the statement of the lemma. If these maps are flat, then $f_\varphi ^*$ is exact (Modules on Sites, Lemma 18.31.2). By definition of the site $\mathcal{C}_{total}$ we see that these functors have the desired exactness properties and we conclude. $\square$

Lemma 84.6.4. In Situation 84.3.3. Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}_{total}$ such that $f_\varphi ^{-1}\mathcal{O}_ n \to \mathcal{O}_ m$ is flat for all $\varphi : [n] \to [m]$. If $\mathcal{I}$ is injective in $\textit{Mod}(\mathcal{O})$, then $\mathcal{I}_ n$ is injective in $\textit{Mod}(\mathcal{O}_ n)$.

Proof. This follows from Homology, Lemma 12.29.1 and Lemma 84.6.3. $\square$

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