The Stacks project

Lemma 34.3.16. Let $\mathit{Sch}_{Zar}$ be a big Zariski site. Let $f : T \to S$ be a morphism in $\mathit{Sch}_{Zar}$.

  1. We have $i_ f = f_{big} \circ i_ T$ with $i_ f$ as in Lemma 34.3.12 and $i_ T$ as in Lemma 34.3.13.

  2. The functor $S_{Zar} \to T_{Zar}$, $(U \to S) \mapsto (U \times _ S T \to T)$ is continuous and induces a morphism of topoi

    \[ f_{small} : \mathop{\mathit{Sh}}\nolimits (T_{Zar}) \longrightarrow \mathop{\mathit{Sh}}\nolimits (S_{Zar}). \]

    The functors $f_{small}^{-1}$ and $f_{small, *}$ agree with the usual notions $f^{-1}$ and $f_*$ is we identify sheaves on $T_{Zar}$, resp. $S_{Zar}$ with sheaves on $T$, resp. $S$ via Lemma 34.3.11.

  3. We have a commutative diagram of morphisms of sites

    \[ \xymatrix{ T_{Zar} \ar[d]_{f_{small}} & (\mathit{Sch}/T)_{Zar} \ar[d]^{f_{big}} \ar[l]^{\pi _ T} \\ S_{Zar} & (\mathit{Sch}/S)_{Zar} \ar[l]_{\pi _ S} } \]

    so that $f_{small} \circ \pi _ T = \pi _ S \circ f_{big}$ as morphisms of topoi.

  4. We have $f_{small} = \pi _ S \circ f_{big} \circ i_ T = \pi _ S \circ i_ f$.

Proof. The equality $i_ f = f_{big} \circ i_ T$ follows from the equality $i_ f^{-1} = i_ T^{-1} \circ f_{big}^{-1}$ which is clear from the descriptions of these functors above. Thus we see (1).

Statement (2): See Sites, Example 7.14.2.

Part (3) follows because $\pi _ S$ and $\pi _ T$ are given by the inclusion functors and $f_{small}$ and $f_{big}$ by the base change functor $U \mapsto U \times _ S T$.

Statement (4) follows from (3) by precomposing with $i_ T$. $\square$


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