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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