The Stacks project

Remark 84.16.5 (Ringed variant). Let $\mathcal{C}$ be a site. Let $\mathcal{O}_\mathcal {C}$ be a sheaf of rings. Given a simplicial semi-representable object $K$ of $\mathcal{C}$ we set $\mathcal{O} = a^{-1}\mathcal{O}_\mathcal {C}$, where $a$ is as in Lemmas 84.16.1 and 84.16.2. The constructions above, keeping track of the sheaves of rings as in Remark 84.15.6, give

1. a ringed site $((\mathcal{C}/K)_{total}, \mathcal{O})$ for a simplicial $K$ object of $\text{SR}(\mathcal{C})$,

2. a morphism of ringed topoi $a : (\mathop{\mathit{Sh}}\nolimits ((\mathcal{C}/K)_{total}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C})$,

3. morphisms of ringed topoi $a_ n : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K_ n), \mathcal{O}_ n) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C})$,

4. a functor $a_! : \textit{Mod}(\mathcal{O}) \to \textit{Mod}(\mathcal{O}_\mathcal {C})$ left adjoint to $a^*$.

The functor $a_!$ exists (but in general is not exact) because $a^{-1}\mathcal{O}_\mathcal {C} = \mathcal{O}$ and we can replace the use of Modules on Sites, Lemma 18.16.2 in the proof of Lemma 84.16.2 by Modules on Sites, Lemma 18.41.1. As discussed in Remark 84.15.6 there are exact functors $a_{n!} : \textit{Mod}(\mathcal{O}_ n) \to \textit{Mod}(\mathcal{O}_\mathcal {C})$ left adjoint to $a_ n^*$. Consequently, the morphisms $a$ and $a_ n$ are flat. Remark 84.15.6 implies the morphism of ringed topoi $f_\varphi : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K_ n), \mathcal{O}_ n) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K_ m), \mathcal{O}_ m)$ for $\varphi : [m] \to [n]$ is flat and there exists an exact functor $f_{\varphi !} : \textit{Mod}(\mathcal{O}_ n) \to \textit{Mod}(\mathcal{O}_ m)$ left adjoint to $f_\varphi ^*$. This in turn implies that for the flat morphism of ringed topoi $g_ n : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K_ n), \mathcal{O}_ n) \to (\mathop{\mathit{Sh}}\nolimits ((\mathcal{C}/K)_{total}), \mathcal{O})$ the functor $g_{n!} : \textit{Mod}(\mathcal{O}_ n) \to \textit{Mod}(\mathcal{O})$ left adjoint to $g_ n^*$ is exact, see Lemma 84.6.3.