The Stacks project

Lemma 34.4.16. Let $\mathit{Sch}_{\acute{e}tale}$ be a big ├ętale site. Let $f : T \to S$ be a morphism in $\mathit{Sch}_{\acute{e}tale}$.

  1. We have $i_ f = f_{big} \circ i_ T$ with $i_ f$ as in Lemma 34.4.12 and $i_ T$ as in Lemma 34.4.13.

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

    \[ f_{small} : T_{\acute{e}tale}\longrightarrow S_{\acute{e}tale} \]

    We have $f_{small, *}(\mathcal{F})(U/S) = \mathcal{F}(U \times _ S T/T)$.

  3. We have a commutative diagram of morphisms of sites

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

    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).

The functor $u : S_{\acute{e}tale}\to T_{\acute{e}tale}$, $u(U \to S) = (U \times _ S T \to T)$ transforms coverings into coverings and commutes with fibre products, see Lemma 34.4.3 (3) and 34.4.10. Moreover, both $S_{\acute{e}tale}$, $T_{\acute{e}tale}$ have final objects, namely $S/S$ and $T/T$ and $u(S/S) = T/T$. Hence by Sites, Proposition 7.14.7 the functor $u$ corresponds to a morphism of sites $T_{\acute{e}tale}\to S_{\acute{e}tale}$. This in turn gives rise to the morphism of topoi, see Sites, Lemma 7.15.2. The description of the pushforward is clear from these references.

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 functors $U \mapsto U \times _ S T$.

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


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