The Stacks project

Lemma 58.22.2. In Situation 58.19.1 assume

  1. $A$ has a dualizing complex and is $f$-adically complete,

  2. every irreducible component of $X$ not contained in $X_0$ has dimension $\geq 3$.

Then the restriction functor

\[ \mathop{\mathrm{colim}}\nolimits _{U_0 \subset U' \subset U\text{ open}} \textit{FÉt}_{U'} \longrightarrow \textit{FÉt}_{U_0} \]

is fully faithful.

Proof. To prove this we may replace $A$ by its reduction by the topological invariance of the fundamental group, see Lemma 58.8.3. Then the result follows from Lemma 58.17.3 and Algebraic and Formal Geometry, Lemma 52.15.7. $\square$

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