The Stacks project

Lemma 58.23.1. In Situation 58.19.1 assume

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

  2. one of the following is true

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

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

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 an equivalence.

Proof. This follows from Lemma 58.17.4 and Algebraic and Formal Geometry, Lemma 52.24.1. $\square$

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