58.25 Purity in local case, II
This section is the continuation of Section 58.20. Recall that we say purity holds for a Noetherian local ring $(A, \mathfrak m)$ if the restriction functor $\textit{FÉt}_ X \to \textit{FÉt}_ U$ is essentially surjective where $X = \mathop{\mathrm{Spec}}(A)$ and $U = X \setminus \{ \mathfrak m\} $.
Lemma 58.25.1. Let $(A, \mathfrak m)$ be a Noetherian local ring. Let $f \in \mathfrak m$. Assume
$A$ has a dualizing complex and is $f$-adically complete,
one of the following is true
$A_ f$ is $(S_2)$ and every irreducible component of $X$ not contained in $X_0$ has dimension $\geq 4$, or
if $\mathfrak p \not\in V(f)$ and $V(\mathfrak p) \cap V(f) \not= \{ \mathfrak m\} $, then $\text{depth}(A_\mathfrak p) + \dim (A/\mathfrak p) > 3$.
for every maximal ideal $\mathfrak p \subset A_ f$ purity holds for $(A_ f)_\mathfrak p$, and
purity holds for $A$.
Then purity holds for $A/fA$.
Proof.
Denote $X = \mathop{\mathrm{Spec}}(A)$ and $U = X \setminus \{ \mathfrak m\} $ the punctured spectrum. Similarly we have $X_0 = \mathop{\mathrm{Spec}}(A/fA)$ and $U_0 = X_0 \setminus \{ \mathfrak m\} $. Let $V_0 \to U_0$ be a finite étale morphism. By Lemma 58.24.1 we find a finite étale morphism $V \to U$ whose base change to $U_0$ is isomorphic to $V_0 \to U_0$. By assumption (5) we find that $V \to U$ extends to a finite étale morphism $Y \to X$. Then the restriction of $Y$ to $X_0$ is the desired extension of $V_0 \to U_0$.
$\square$
Lemma 58.25.2. Let $(A, \mathfrak m)$ be a Noetherian local ring. Let $f \in \mathfrak m$. Assume
$A$ is $f$-adically complete,
$f$ is a nonzerodivisor,
$H^1_\mathfrak m(A/fA)$ and $H^2_\mathfrak m(A/fA)$ are finite $A$-modules,
for every maximal ideal $\mathfrak p \subset A_ f$ purity holds for $(A_ f)_\mathfrak p$,
purity holds for $A$.
Then purity holds for $A/fA$.
Proof.
The proof is identical to the proof of Lemma 58.25.1 using Lemma 58.24.2 in stead of Lemma 58.24.1.
$\square$
Now we can bootstrap the earlier results to prove that purity holds for complete intersections of dimension $\geq 3$. Recall that a Noetherian local ring is called a complete intersection if its completion is the quotient of a regular local ring by the ideal generated by a regular sequence. See the discussion in Divided Power Algebra, Section 23.8.
Proposition 58.25.3. Let $(A, \mathfrak m)$ be a Noetherian local ring. If $A$ is a complete intersection of dimension $\geq 3$, then purity holds for $A$ in the sense that any finite étale cover of the punctured spectrum extends.
Proof.
By Lemma 58.20.4 we may assume that $A$ is a complete local ring. By assumption we can write $A = B/(f_1, \ldots , f_ r)$ where $B$ is a complete regular local ring and $f_1, \ldots , f_ r$ is a regular sequence. We will finish the proof by induction on $r$. The base case is $r = 0$ which follows from Lemma 58.21.3 which applies to regular rings of dimension $\geq 2$.
Assume that $A = B/(f_1, \ldots , f_ r)$ and that the proposition holds for $r - 1$. Set $A' = B/(f_1, \ldots , f_{r - 1})$ and apply Lemma 58.25.2 to $f_ r \in A'$. This is permissible: condition (1) holds as $f_1, \ldots , f_ r$ is a regular sequence, condition (2) holds as $B$ and hence $A'$ is complete, condition (3) holds as $A = A'/f_ r A'$ is Cohen-Macaulay of dimension $\dim (A) \geq 3$, see Dualizing Complexes, Lemma 47.11.1, condition (4) holds by induction hypothesis as $\dim ((A'_{f_ r})_\mathfrak p) \geq 3$ for a maximal prime $\mathfrak p$ of $A'_{f_ r}$ and as $(A'_{f_ r})_\mathfrak p = B_\mathfrak q/(f_1, \ldots , f_{r - 1})$ for some $\mathfrak q \subset B$, condition (5) holds by induction hypothesis.
$\square$
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