Remark 59.75.1 (Alternative description of sp). Let $S$, $\overline{s}$, and $\overline{t}$ be as above. Another way to describe the specialization map is to use that
\[ \mathcal{F}_{\overline{s}} = \Gamma (\mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{s}}), p^{-1}\mathcal{F}) \quad \text{and}\quad \mathcal{F}_{\overline{t}} = \Gamma (\overline{t}, \overline{t}^{-1}p^{-1}\mathcal{F}) \]
The first equality follows from Theorem 59.53.1 applied to $\text{id}_ S : S \to S$ and the second equality follows from Lemma 59.36.2. Then we can think of $sp$ as the map
\[ sp : \mathcal{F}_{\overline{s}} = \Gamma (\mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{s}}), p^{-1}\mathcal{F}) \xrightarrow {\text{pullback by }\overline{t}} \Gamma (\overline{t}, \overline{t}^{-1}p^{-1}\mathcal{F}) = \mathcal{F}_{\overline{t}} \]
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