The Stacks project

Remark 59.75.8. Let $f : X \to S$ be a morphism of schemes. Let $K \in D(X_{\acute{e}tale})$. Let $\overline{s}$ be a geometric point of $S$ and let $\overline{t}$ be a geometric point of $\mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{s}})$. Let $c$ be as in Remark 59.75.2. We can always make a commutative diagram

\[ \xymatrix{ (Rf_*K)_{\overline{s}} \ar[r] \ar[d]_{sp} & R\Gamma (X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S, \overline{s}}^{sh}), K) \ar[r] \ar[d]_{(\text{id}_ X \times c)^{-1}} & R\Gamma (X_{\overline{s}}, K) \\ (Rf_*K)_{\overline{t}} \ar[r] & R\Gamma (X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S, \overline{t}}^{sh}), K) \ar[r] & R\Gamma (X_{\overline{t}}, K) } \]

where the horizontal arrows are those of Remark 59.53.2. In general there won't be a vertical map on the right between the cohomologies of $K$ on the fibres fitting into this diagram, even in the case of Lemma 59.75.7.

Comments (0)

There are also:

  • 2 comment(s) on Section 59.75: Specializations and étale sheaves

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Your email address will not be published.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 0GJ9. Beware of the difference between the letter 'O' and the digit '0'.