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59.37 Functoriality of big topoi

Given a morphism of schemes $f : X \to Y$ there are a whole host of morphisms of topoi associated to $f$, see Topologies, Section 34.11 for a list. Perhaps the most used ones are the morphisms of topoi

\[ f_{big} = f_{big, \tau } : \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/X)_\tau ) \longrightarrow \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/Y)_\tau ) \]

where $\tau \in \{ Zariski, {\acute{e}tale}, smooth, syntomic, fppf\} $. These each correspond to a continuous functor

\[ (\mathit{Sch}/Y)_\tau \longrightarrow (\mathit{Sch}/X)_\tau , \quad V/Y \longmapsto X \times _ Y V/X \]

which preserves final objects, fibre products and covering, and hence defines a morphism of sites

\[ f_{big} : (\mathit{Sch}/X)_\tau \longrightarrow (\mathit{Sch}/Y)_\tau . \]

See Topologies, Sections 34.3, 34.4, 34.5, 34.6, and 34.7. In particular, pushforward along $f_{big}$ is given by the rule

\[ (f_{big, *}\mathcal{F})(V/Y) = \mathcal{F}(X \times _ Y V/X) \]

It turns out that these morphisms of topoi have an inverse image functor $f_{big}^{-1}$ which is very easy to describe. Namely, we have

\[ (f_{big}^{-1}\mathcal{G})(U/X) = \mathcal{G}(U/Y) \]

where the structure morphism of $U/Y$ is the composition of the structure morphism $U \to X$ with $f$, see Topologies, Lemmas 34.3.15, 34.4.15, 34.5.10, 34.6.10, and 34.7.12.


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