The Stacks project

Lemma 34.4.17. Let $\mathit{Sch}_{\acute{e}tale}$ be a big étale 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 34.4.13 and $i_ T$ as in Lemma 34.4.14.

  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 $i_ T$. $\square$

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

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.

All contributions are licensed under the GNU Free Documentation License.




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 021I. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 34.4.17 as pdf