The Stacks project

Lemma 58.19.5. In Situation 58.19.1. Assume

  1. $A$ has a dualizing complex,

  2. the pair $(A, (f))$ is henselian,

  3. 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 $\geq 3$, or

    2. for every prime $\mathfrak p \subset A$, $f \not\in \mathfrak p$ we have $\text{depth}(A_\mathfrak p) + \dim (A/\mathfrak p) > 2$.

Then the restriction functor $\textit{FÉt}_ U \longrightarrow \textit{FÉt}_{U_0}$ is fully faithful.

Proof. Let $A'$ be the $\mathfrak m$-adic completion of $A$. We will show that the hypotheses remain true for $A'$. This is clear for conditions (a) and (b). Condition (c)(ii) is preserved by Local Cohomology, Lemma 51.11.3. Next, assume (c)(i) holds. Since $A$ is universally catenary (Dualizing Complexes, Lemma 47.17.4) we see that every irreducible component of $\mathop{\mathrm{Spec}}(A')$ not contained in $V(f)$ has dimension $\geq 3$, see More on Algebra, Proposition 15.109.5. Since $A \to A'$ is flat with Gorenstein fibres, the condition that $A_ f$ is $(S_2)$ implies that $A'_ f$ is $(S_2)$. References used: Dualizing Complexes, Section 47.23, More on Algebra, Section 15.51, and Algebra, Lemma 10.163.4. Thus by Lemma 58.19.4 we may assume that $A$ is a Noetherian complete local ring.

Assume $A$ is a complete local ring in addition to the other assumptions. By Lemma 58.17.1 the result follows from Algebraic and Formal Geometry, Lemma 52.15.6. $\square$

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