58.26 Purity in local case, III

In this section is a continuation of the discussion in Sections 58.20 and 58.25.

Lemma 58.26.1. Let $(A, \mathfrak m)$ be a Noetherian local ring of depth $\geq 2$. Let $B = A[[x_1, \ldots , x_ d]]$ with $d \geq 1$. Set $Y = \mathop{\mathrm{Spec}}(B)$ and $Y_0 = V(x_1, \ldots , x_ d)$. For any open subscheme $V \subset Y$ with $V_0 = V \cap Y_0$ equal to $Y_0 \setminus \{ \mathfrak m_ B\}$ the restriction functor

$\textit{FÉt}_ V \longrightarrow \textit{FÉt}_{V_0}$

is fully faithful.

Proof. Set $I = (x_1, \ldots , x_ d)$. Set $X = \mathop{\mathrm{Spec}}(A)$. If we use the map $Y \to X$ to identify $Y_0$ with $X$, then $V_0$ is identified with the punctured spectrum $U$ of $A$. Pushing forward modules by this affine morphism we get

\begin{align*} \mathop{\mathrm{lim}}\nolimits _ n \Gamma (V_0, \mathcal{O}_ V/I^ n\mathcal{O}_ V) & = \mathop{\mathrm{lim}}\nolimits _ n \Gamma (V_0, \mathcal{O}_ Y/I^ n\mathcal{O}_ Y) \\ & = \mathop{\mathrm{lim}}\nolimits _ n \Gamma (U, \mathcal{O}_ U[x_1, \ldots , x_ d]/(x_1, \ldots , x_ d)^ n) \\ & = \mathop{\mathrm{lim}}\nolimits _ n A[x_1, \ldots , x_ d]/(x_1, \ldots , x_ d)^ n \\ & = B \end{align*}

Namely, as the depth of $A$ is $\geq 2$ we have $\Gamma (U, \mathcal{O}_ U) = A$, see Local Cohomology, Lemma 51.8.2. Thus for any $V \subset Y$ open as in the lemma we get

$B = \Gamma (Y, \mathcal{O}_ Y) \to \Gamma (V, \mathcal{O}_ V) \to \mathop{\mathrm{lim}}\nolimits _ n \Gamma (V_0, \mathcal{O}_ Y/I^ n\mathcal{O}_ Y) = B$

which implies both arrows are isomorphisms (small detail omitted). By Algebraic and Formal Geometry, Lemma 52.15.1 we conclude that $\textit{Coh}(\mathcal{O}_ V) \to \textit{Coh}(V, I\mathcal{O}_ V)$ is fully faithful on the full subcategory of finite locally free objects. Thus we conclude by Lemma 58.17.1. $\square$

Lemma 58.26.2. Let $(A, \mathfrak m)$ be a Noetherian local ring of depth $\geq 2$. Let $B = A[[x_1, \ldots , x_ d]]$ with $d \geq 1$. For any open $V \subset Y = \mathop{\mathrm{Spec}}(B)$ which contains

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

2. the prime $\mathfrak m B$

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 $\mathfrak m$. In this case $A \to B_\mathfrak q$ is a flat local homomorphism and hence $\text{depth}(B_\mathfrak q) \geq 2$ (Algebra, Lemma 10.163.2). Thus the functor is fully faithful by Lemma 58.10.3 combined with Local Cohomology, Lemma 51.3.1.

Let $W \to V$ be a finite étale morphism. Let $B \to C$ be the unique finite ring map such that $\mathop{\mathrm{Spec}}(C) \to Y$ is the finite morphism extending $W \to V$ constructed in Lemma 58.21.5. Observe that $C = \Gamma (W, \mathcal{O}_ W)$.

Set $Y_0 = V(x_1, \ldots , x_ d)$ and $V_0 = V \cap Y_0$. Set $X = \mathop{\mathrm{Spec}}(A)$. If we use the map $Y \to X$ to identify $Y_0$ with $X$, then $V_0$ is identified with the punctured spectrum $U$ of $A$. Thus we may view $W_0 = W \times _ Y Y_0$ as a finite étale scheme over $U$. Then

$W_0 \times _ U (U \times _ X Y) \quad \text{and}\quad W \times _ V (U \times _ X Y)$

are schemes finite étale over $U \times _ X Y$ which restrict to isomorphic finite étale schemes over $V_0$. By Lemma 58.26.1 applied to the open $U \times _ X Y$ we obtain an isomorphism

$W_0 \times _ U (U \times _ X Y) \longrightarrow W \times _ V (U \times _ X Y)$

over $U \times _ X Y$.

Observe that $C_0 = \Gamma (W_0, \mathcal{O}_{W_0})$ is a finite $A$-algebra by Lemma 58.21.5 applied to $W_0 \to U \subset X$ (exactly as we did for $B \to C$ above). Since the construction in Lemma 58.21.5 is compatible with flat base change and with change of opens, the isomorphism above induces an isomorphism

$\Psi : C \longrightarrow C_0 \otimes _ A B$

of finite $B$-algebras. However, we know that $\mathop{\mathrm{Spec}}(C) \to Y$ is étale at all points above at least one point of $Y$ lying over $\mathfrak m \in X$. Since $\Psi$ is an isomorphism, we conclude that $\mathop{\mathrm{Spec}}(C_0) \to X$ is étale above $\mathfrak m$ (small detail omitted). Of course this means that $A \to C_0$ is finite étale and hence $B \to C$ is finite étale. $\square$

Lemma 58.26.3. Let $f : X \to S$ be a morphism of schemes. Let $U \subset X$ be an open subscheme. Assume

1. $f$ is smooth,

2. $S$ is Noetherian,

3. for $s \in S$ with $\text{depth}(\mathcal{O}_{S, s}) \leq 1$ we have $X_ s = U_ s$,

4. $U_ s \subset X_ s$ is dense for all $s \in S$.

Then $\textit{FÉt}_ X \to \textit{FÉt}_ U$ is an equivalence.

Proof. The functor is fully faithful by Lemma 58.10.3 combined with Local Cohomology, Lemma 51.3.1 (plus an application of Algebra, Lemma 10.163.2 to check the depth condition).

Let $\pi : V \to U$ be a finite étale morphism. Let $Y \to X$ be the finite morphism constructed in Lemma 58.21.5. We have to show that $Y \to X$ is finite étale. To show that this is true for all points $x \in X$ mapping to a given point $s \in S$ we may perform a base change by a flat morphism $S' \to S$ of Noetherian schemes such that $s$ is in the image. This follows from the compatibility of the construction in Lemma 58.21.5 with flat base change.

After enlarging $U$ we may assume $U \subset X$ is the maximal open over which $Y \to X$ is finite étale. Let $Z \subset X$ be the complement of $U$. To get a contradiction, assume $Z \not= \emptyset$. Let $s \in S$ be a point in the image of $Z \to S$ such that no strict generalization of $s$ is in the image. Then after base change to $\mathop{\mathrm{Spec}}(\mathcal{O}_{S, s})$ we see that $S = \mathop{\mathrm{Spec}}(A)$ with $(A, \mathfrak m, \kappa )$ a local Noetherian ring of depth $\geq 2$ and $Z$ contained in the closed fibre $X_ s$ and nowhere dense in $X_ s$. Choose a closed point $z \in Z$. Then $\kappa (z)/\kappa$ is finite (by the Hilbert Nullstellensatz, see Algebra, Theorem 10.34.1). Choose a finite flat morphism $(S', s') \to (S, s)$ of local schemes realizing the residue field extension $\kappa (z)/\kappa$, see Algebra, Lemma 10.159.3. After doing a base change by $S' \to S$ we reduce to the case where $\kappa (z) = \kappa$.

By More on Morphisms, Lemma 37.38.5 there exists a locally closed subscheme $S' \subset X$ passing through $z$ such that $S' \to S$ is étale at $z$. After performing the base change by $S' \to S$, we may assume there is a section $\sigma : S \to X$ such that $\sigma (s) = z$. Choose an affine neighbourhood $\mathop{\mathrm{Spec}}(B) \subset X$ of $s$. Then $A \to B$ is a smooth ring map which has a section $\sigma : B \to A$. Denote $I = \mathop{\mathrm{Ker}}(\sigma )$ and denote $B^\wedge$ the $I$-adic completion of $B$. Then $B^\wedge \cong A[[x_1, \ldots , x_ d]]$ for some $d \geq 0$, see Algebra, Lemma 10.139.4. Observe that $d > 0$ since otherwise we see that $X \to S$ is étale at $z$ which would imply that $z$ is a generic point of $X_ s$ and hence $z \in U$ by assumption (4). Similarly, if $d > 0$, then $\mathfrak m B^\wedge$ maps into $U$ via the morphism $\mathop{\mathrm{Spec}}(B^\wedge ) \to X$. It suffices prove $Y \to X$ is finite étale after base change to $\mathop{\mathrm{Spec}}(B^\wedge )$. Since $B \to B^\wedge$ is flat (Algebra, Lemma 10.97.2) this follows from Lemma 58.26.2 and the uniqueness in the construction of $Y \to X$. $\square$

Proposition 58.26.4. 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 10.163.2). Thus the functor is fully faithful by Lemma 58.10.3 combined with Local Cohomology, Lemma 51.3.1.

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 15.43.1, 15.43.2, and 15.37.4. By the uniqueness and compatibility with flat base change of the construction of Lemma 58.21.5 it suffices to prove the essential surjectivity for $A^\wedge \to B^\wedge$ and the inverse image of $V$ (details omitted; compare with Lemma 58.20.4 for the case where $V$ is the punctured spectrum). By More on Algebra, Proposition 15.49.2 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 32.4.11). 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 32.10.1, 32.8.3, and 32.8.10. 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 58.26.3 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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