The Stacks project

Lemma 59.87.1. Consider the cartesian diagram of schemes

\[ \xymatrix{ X \ar[d]_ f & Y \ar[l]^ h \ar[d]^ e \\ S & T \ar[l]_ g } \]

Assume that $f$ is flat and every object $U$ of $X_{\acute{e}tale}$ has a covering $\{ U_ i \to U\} $ such that $U_ i \to S$ factors as $U_ i \to V_ i \to S$ with $V_ i \to S$ étale and $U_ i \to V_ i$ quasi-compact with geometrically connected fibres. Then for any sheaf $\mathcal{F}$ of sets on $T_{\acute{e}tale}$ we have $f^{-1}g_*\mathcal{F} = h_*e^{-1}\mathcal{F}$.

Proof. Let $U \to X$ be an étale morphism such that $U \to S$ factors as $U \to V \to S$ with $V \to S$ étale and $U \to V$ quasi-compact with geometrically connected fibres. Observe that $U \to V$ is flat (More on Flatness, Lemma 38.2.3). We claim that

\begin{align*} f^{-1}g_*\mathcal{F}(U) & = g_*\mathcal{F}(V) \\ & = \mathcal{F}(V \times _ S T) \\ & = e^{-1}\mathcal{F}(U \times _ X Y) \\ & = h_*e^{-1}\mathcal{F}(U) \end{align*}

Namely, thinking of $U$ as an object of $X_{\acute{e}tale}$ and $V$ as an object of $S_{\acute{e}tale}$ we see that the first equality follows from Lemma 59.39.31. Thinking of $V \times _ S T$ as an object of $T_{\acute{e}tale}$ the second equality follows from the definition of $g_*$. Observe that $U \times _ X Y = U \times _ S T$ (because $Y = X \times _ S T$) and hence $U \times _ X Y \to V \times _ S T$ has geometrically connected fibres as a base change of $U \to V$. Thinking of $U \times _ X Y$ as an object of $Y_{\acute{e}tale}$, we see that the third equality follows from Lemma 59.39.3 as before. Finally, the fourth equality follows from the definition of $h_*$.

Since by assumption every object of $X_{\acute{e}tale}$ has an étale covering to which the argument of the previous paragraph applies we see that the lemma is true. $\square$

[1] Strictly speaking, we are also using that the restriction of $f^{-1}g_*\mathcal{F}$ to $U_{\acute{e}tale}$ is the pullback via $U \to V$ of the restriction of $g_*\mathcal{F}$ to $V_{\acute{e}tale}$. See Sites, Lemma 7.28.2.

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Your email address will not be published.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 0EZX. Beware of the difference between the letter 'O' and the digit '0'.