Lemma 61.19.3. Let $X$ be a scheme. Let $Y = \mathop{\mathrm{lim}}\nolimits Y_ i$ be the limit of a directed inverse system of quasi-compact and quasi-separated objects of $X_{pro\text{-}\acute{e}tale}$ with affine transition morphisms. For any sheaf $\mathcal{F}$ on $X_{\acute{e}tale}$ we have

\[ \epsilon ^{-1}\mathcal{F}(Y) = \mathop{\mathrm{colim}}\nolimits \epsilon ^{-1}\mathcal{F}(Y_ i) \]

Moreover, if $Y_ i$ is in $X_{\acute{e}tale}$ we have $\epsilon ^{-1}\mathcal{F}(Y) = \mathop{\mathrm{colim}}\nolimits \mathcal{F}(Y_ i)$.

**Proof.**
By the description of $\epsilon ^{-1}\mathcal{F}$ in Lemma 61.19.1, the displayed formula is a special case of Étale Cohomology, Theorem 59.51.3. (When $X$, $Y$, and the $Y_ i$ are all affine, see the easier to parse Étale Cohomology, Lemma 59.51.5.) The final statement follows immediately from this and Lemma 61.19.2.
$\square$

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