The Stacks project

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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