Lemma 58.24.1. In Situation 58.19.1 assume

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

one of the following is true

$A_ f$ is $(S_2)$ and every irreducible component of $X$ not contained in $X_0$ has dimension $\geq 4$, or

if $\mathfrak p \not\in V(f)$ and $V(\mathfrak p) \cap V(f) \not= \{ \mathfrak m\} $, then $\text{depth}(A_\mathfrak p) + \dim (A/\mathfrak p) > 3$.

for every maximal ideal $\mathfrak p \subset A_ f$ purity holds for $(A_ f)_\mathfrak p$.

Then the restriction functor $\textit{FÉt}_ U \to \textit{FÉt}_{U_0}$ is essentially surjective.

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