**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.
$\square$

## Comments (0)