The Stacks project

Remark 85.16.6 (Ringed variant over an object). Let $\mathcal{C}$ be a site. Let $\mathcal{O}_\mathcal {C}$ be a sheaf of rings. Let $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and denote $\mathcal{O}_ X = \mathcal{O}_\mathcal {C}|_{\mathcal{C}/X}$. Then we can combine the constructions given in Remarks 85.16.4 and 85.16.5 to get

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

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

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

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

Of course, all the results mentioned in Remark 85.16.5 hold in this setting as well.

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