The Stacks project

Proposition 57.25.3. 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 ├ętale cover of the punctured spectrum extends.

Proof. By Lemma 57.20.4 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 57.21.3 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 57.25.2 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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