Remark 58.23.3. Let (A, \mathfrak m) be a complete local Noetherian ring and f \in \mathfrak m nonzero. Suppose that A_ f is (S_2) and every irreducible component of \mathop{\mathrm{Spec}}(A) has dimension \geq 4. Then Lemma 58.23.1 tells us that the category
\mathop{\mathrm{colim}}\nolimits \nolimits _{U' \subset U\text{ open, }U_0 \subset U} \text{ category of schemes finite étale over }U'
is equivalent to the category of schemes finite étale over U_0. For example this holds if A is a normal domain of dimension \geq 4!
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