The Stacks project

Remark 59.75.2 (Yet another description of sp). Let $S$, $\overline{s}$, and $\overline{t}$ be as above. Another alternative is to use the unique morphism

\[ c : \mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{t}}) \longrightarrow \mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{s}}) \]

over $S$ which is compatible with the given morphism $\overline{t} \to \mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{s}})$ and the morphism $\overline{t} \to \mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{t, \overline{t}})$. The uniqueness and existence of the displayed arrow follows from Algebra, Lemma 10.154.6 applied to $\mathcal{O}_{S, s}$, $\mathcal{O}^{sh}_{S, \overline{t}}$, and $\mathcal{O}^{sh}_{S, \overline{s}} \to \kappa (\overline{t})$. We obtain

\[ sp : \mathcal{F}_{\overline{s}} = \Gamma (\mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{s}}), \mathcal{F}) \xrightarrow {\text{pullback by }c} \Gamma (\mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{t}}), \mathcal{F}) = \mathcal{F}_{\overline{t}} \]

(with obvious notational conventions). In fact this procedure also works for objects $K$ in $D(S_{\acute{e}tale})$: the specialization map for $K$ is equal to the map

\[ sp : K_{\overline{s}} = R\Gamma (\mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{s}}), K) \xrightarrow {\text{pullback by }c} R\Gamma (\mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{t}}), K) = K_{\overline{t}} \]

The equality signs are valid as taking global sections over the striclty henselian schemes $\mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{s}})$ and $\mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{t}})$ is exact (and the same as taking stalks at $\overline{s}$ and $\overline{t}$) and hence no subtleties related to the fact that $K$ may be unbounded arise.


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Your email address will not be published.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 0GJ4. Beware of the difference between the letter 'O' and the digit '0'.