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The Stacks project

Lemma 59.49.1. Let $\tau \in \{ Zariski, {\acute{e}tale}, smooth, syntomic, fppf\} $. Let $f : X \to Y$ be a morphism of schemes. Let

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

be the corresponding morphism of topoi as in Topologies, Lemma 34.3.16, 34.4.16, 34.5.10, 34.6.10, or 34.7.12.

  1. The functor $f_{big}^{-1} : \textit{Ab}((\mathit{Sch}/Y)_\tau ) \to \textit{Ab}((\mathit{Sch}/X)_\tau )$ has a left adjoint

    \[ f_{big!} : \textit{Ab}((\mathit{Sch}/X)_\tau ) \to \textit{Ab}((\mathit{Sch}/Y)_\tau ) \]

    which is exact.

  2. The functor $f_{big}^* : \textit{Mod}((\mathit{Sch}/Y)_\tau , \mathcal{O}) \to \textit{Mod}((\mathit{Sch}/X)_\tau , \mathcal{O})$ has a left adjoint

    \[ f_{big!} : \textit{Mod}((\mathit{Sch}/X)_\tau , \mathcal{O}) \to \textit{Mod}((\mathit{Sch}/Y)_\tau , \mathcal{O}) \]

    which is exact.

Moreover, the two functors $f_{big!}$ agree on underlying sheaves of abelian groups.

Proof. Recall that $f_{big}$ is the morphism of topoi associated to the continuous and cocontinuous functor $u : (\mathit{Sch}/X)_\tau \to (\mathit{Sch}/Y)_\tau $, $U/X \mapsto U/Y$. Moreover, we have $f_{big}^{-1}\mathcal{O} = \mathcal{O}$. Hence the existence of $f_{big!}$ follows from Modules on Sites, Lemma 18.16.2, respectively Modules on Sites, Lemma 18.41.1. Note that if $U$ is an object of $(\mathit{Sch}/X)_\tau $ then the functor $u$ induces an equivalence of categories

\[ u' : (\mathit{Sch}/X)_\tau /U \longrightarrow (\mathit{Sch}/Y)_\tau /U \]

because both sides of the arrow are equal to $(\mathit{Sch}/U)_\tau $. Hence the agreement of $f_{big!}$ on underlying abelian sheaves follows from the discussion in Modules on Sites, Remark 18.41.2. The exactness of $f_{big!}$ follows from Modules on Sites, Lemma 18.16.3 as the functor $u$ above which commutes with fibre products and equalizers. $\square$

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