The Stacks project

Lemma 73.4.11. Let $S$ be a scheme. Let $f : Y \to X$ be a morphism in $(\textit{Spaces}/S)_{\acute{e}tale}$.

  1. We have $i_ f = f_{big} \circ i_ T$ with $i_ f$ as in Lemma 73.4.7 and $i_ T$ as in Lemma 73.4.8.

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

    \[ f_{spaces, {\acute{e}tale}} : Y_{spaces, {\acute{e}tale}} \longrightarrow X_{spaces, {\acute{e}tale}} \]

    The corresponding morphism of small étale topoi is denoted

    \[ f_{small} : \mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}) \]

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

  3. We have a commutative diagram of morphisms of sites

    \[ \xymatrix{ Y_{spaces, {\acute{e}tale}} \ar[d]_{f_{spaces, {\acute{e}tale}}} & (\textit{Spaces}/Y)_{\acute{e}tale}\ar[d]^{f_{big}} \ar[l]^-{\pi _ Y}\\ X_{spaces, {\acute{e}tale}} & (\textit{Spaces}/X)_{\acute{e}tale}\ar[l]_-{\pi _ X} } \]

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

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

Proof. The equality $i_ f = f_{big} \circ i_ Y$ 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 $u : X_{spaces, {\acute{e}tale}} \to Y_{spaces, {\acute{e}tale}}$, $u(U \to X) = (U \times _ X Y \to Y)$ was shown to give rise to a morphism of sites and correspong morphism of small étale topoi in Properties of Spaces, Lemma 66.18.8. The description of the pushforward is clear.

Part (3) follows because $\pi _ X$ and $\pi _ Y$ are given by the inclusion functors and $f_{spaces, {\acute{e}tale}}$ and $f_{big}$ by the base change functors $U \mapsto U \times _ X Y$.

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

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