The Stacks project

Lemma 59.87.4. Consider a cartesian diagram of schemes

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

Assume that

  1. $f$ is flat and open,

  2. the residue fields of $S$ are separably algebraically closed,

  3. given an étale morphism $U \to X$ with $U$ affine we can write $U$ as a finite disjoint union of open subschemes of $X$ (for example if $X$ is a normal integral scheme with separably closed function field),

  4. any nonempty open of a fibre $X_ s$ of $f$ is connected (for example if $X_ s$ is irreducible or empty).

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. Omitted. Hint: the assumptions almost trivially imply the condition of Lemma 59.87.1. The for example in part (3) follows from Lemma 59.80.4. $\square$

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Your email address will not be published.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



View Lemma 59.87.4 as pdf