
Proposition 53.26.4. Let $A \to B$ be a local homomorphism of local Noetherian rings. Assume $A$ has depth $\geq 2$, $A \to B$ is formally smooth for the $\mathfrak m_ B$-adic topology, and $\dim (B) > \dim (A)$. For any open $V \subset Y = \mathop{\mathrm{Spec}}(B)$ which contains

1. any prime $\mathfrak q \subset B$ such that $\mathfrak q \cap A \not= \mathfrak m_ A$,

2. the prime $\mathfrak m_ A B$

the functor $\textit{FÉt}_ Y \to \textit{FÉt}_ V$ is an equivalence. In particular purity holds for $B$.

Proof. A prime $\mathfrak q \subset B$ which is not contained in $V$ lies over $\mathfrak m_ A$. In this case $A \to B_\mathfrak q$ is a flat local homomorphism and hence $\text{depth}(B_\mathfrak q) \geq 2$ (Algebra, Lemma 10.157.2). Thus the functor is fully faithful by Lemma 53.10.4 combined with Lemma 53.10.3.

Denote $A^\wedge$ and $B^\wedge$ the completions of $A$ and $B$ with respect to their maximal ideals. Observe that the assumptions of the proposition hold for $A^\wedge \to B^\wedge$, see More on Algebra, Lemmas 15.42.1, 15.42.2, and 15.36.4. By the uniqueness and compatibility with flat base change of the construction of Lemma 53.21.5 it suffices to prove the essential surjectivity for $A^\wedge \to B^\wedge$ and the inverse image of $V$ (details omitted; compare with Lemma 53.20.4 for the case where $V$ is the punctured spectrum). By More on Algebra, Proposition 15.48.2 this means we may assume $A \to B$ is regular.

Let $W \to V$ be a finite étale morphism. By Popescu's theorem (Smoothing Ring Maps, Theorem 16.12.1) we can write $B = \mathop{\mathrm{colim}}\nolimits B_ i$ as a filtered colimit of smooth $A$-algebras. We can pick an $i$ and an open $V_ i \subset \mathop{\mathrm{Spec}}(B_ i)$ whose inverse image is $V$ (Limits, Lemma 31.4.11). After increasing $i$ we may assume there is a finite étale morphism $W_ i \to V_ i$ whose base change to $V$ is $W \to V$, see Limits, Lemmas 31.10.1, 31.8.3, and 31.8.10. We may assume the complement of $V_ i$ is contained in the closed fibre of $\mathop{\mathrm{Spec}}(B_ i) \to \mathop{\mathrm{Spec}}(A)$ as this is true for $V$ (either choose $V_ i$ this way or use the lemma above to show this is true for $i$ large enough). Let $\eta$ be the generic point of the closed fibre of $\mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(A)$. Since $\eta \in V$, the image of $\eta$ is in $V_ i$. Hence after replacing $V_ i$ by an affine open neighbourhood of the image of the closed point of $\mathop{\mathrm{Spec}}(B)$, we may assume that the closed fibre of $\mathop{\mathrm{Spec}}(B_ i) \to \mathop{\mathrm{Spec}}(A)$ is irreducible and that its generic point is contained in $V_ i$ (details omitted; use that a scheme smooth over a field is a disjoint union of irreducible schemes). At this point we may apply Lemma 53.26.3 to see that $W_ i \to V_ i$ extends to a finite étale morphism $\mathop{\mathrm{Spec}}(C_ i) \to \mathop{\mathrm{Spec}}(B_ i)$ and pulling back to $\mathop{\mathrm{Spec}}(B)$ we conclude that $W$ is in the essential image of the functor $\textit{FÉt}_ Y \to \textit{FÉt}_ V$ as desired. $\square$

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