The Stacks project

Remark 35.8.4. Let $f : T \to S$ be a morphism of schemes. Each of the morphisms of sites $f_{sites}$ listed in Topologies, Section 34.11 becomes a morphism of ringed sites. Namely, each of these morphisms of sites $f_{sites} : (\mathit{Sch}/T)_\tau \to (\mathit{Sch}/S)_{\tau '}$, or $f_{sites} : (\mathit{Sch}/S)_\tau \to S_{\tau '}$ is given by the continuous functor $S'/S \mapsto T \times _ S S'/S$. Hence, given $S'/S$ we let

be the usual map $\text{pr}_{S'}^\sharp : \mathcal{O}(S') \to \mathcal{O}(T \times _ S S')$. Similarly, the morphism $i_ f : \mathop{\mathit{Sh}}\nolimits (T_\tau ) \to \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_\tau )$ for $\tau \in \{ Zar, {\acute{e}tale}\}$, see Topologies, Lemmas 34.3.13 and 34.4.13, becomes a morphism of ringed topoi because $i_ f^{-1}\mathcal{O} = \mathcal{O}$. Here are some special cases:

1. The morphism of big sites $f_{big} : (\mathit{Sch}/X)_{fppf} \to (\mathit{Sch}/Y)_{fppf}$, becomes a morphism of ringed sites

   $$(f_{big}, f_{big}^\sharp ) : ((\mathit{Sch}/X)_{fppf}, \mathcal{O}_ X) \longrightarrow ((\mathit{Sch}/Y)_{fppf}, \mathcal{O}_ Y)$$

   as in Modules on Sites, Definition 18.6.1. Similarly for the big syntomic, smooth, étale and Zariski sites.

2. The morphism of small sites $f_{small} : X_{\acute{e}tale}\to Y_{\acute{e}tale}$ becomes a morphism of ringed sites

   $$(f_{small}, f_{small}^\sharp ) : (X_{\acute{e}tale}, \mathcal{O}_ X) \longrightarrow (Y_{\acute{e}tale}, \mathcal{O}_ Y)$$

   as in Modules on Sites, Definition 18.6.1. Similarly for the small Zariski site.