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59.34 Functoriality of small étale topos

So far we haven't yet discussed the functoriality of the étale site, in other words what happens when given a morphism of schemes. A precise formal discussion can be found in Topologies, Section 34.4. In this and the next sections we discuss this material briefly specifically in the setting of small étale sites.

Let $f : X \to Y$ be a morphism of schemes. We obtain a functor
\begin{equation} \label{etale-cohomology-equation-functorial} u : Y_{\acute{e}tale}\longrightarrow X_{\acute{e}tale}, \quad V/Y \longmapsto X \times _ Y V/X. \end{equation}

This functor has the following important properties

  1. $u(\text{final object}) = \text{final object}$,

  2. $u$ preserves fibre products,

  3. if $\{ V_ j \to V\} $ is a covering in $Y_{\acute{e}tale}$, then $\{ u(V_ j) \to u(V)\} $ is a covering in $X_{\acute{e}tale}$.

Each of these is easy to check (omitted). As a consequence we obtain what is called a morphism of sites

\[ f_{small} : X_{\acute{e}tale}\longrightarrow Y_{\acute{e}tale}, \]

see Sites, Definition 7.14.1 and Sites, Proposition 7.14.7. It is not necessary to know about the abstract notion in detail in order to work with étale sheaves and étale cohomology. It usually suffices to know that there are functors $f_{small, *}$ (pushforward) and $f_{small}^{-1}$ (pullback) on étale sheaves, and to know some of their simple properties. We will discuss these properties in the next sections, but we will sometimes refer to the more abstract material for proofs since that is often the natural setting to prove them.

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