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 $i_ T$. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0211. Beware of the difference between the letter 'O' and the digit '0'.