The Stacks project

Proposition 52.27.6 (Grothendieck). Let $(A, \mathfrak m)$ be a Noetherian local ring. If $A$ is a complete intersection of dimension $\geq 4$, then the Picard group of the punctured spectrum of $A$ is trivial.

Proof. By Lemma 52.27.4 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 52.27.5.

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 52.27.3 to $f_ r \in A'$. This is permissible:

  1. condition (1) of Lemma 52.27.1 holds because our local rings are complete,

  2. condition (2) of Lemma 52.27.1 holds holds as $f_1, \ldots , f_ r$ is a regular sequence,

  3. condition (3) and (4) of Lemma 52.27.1 hold as $A = A'/f_ r A'$ is Cohen-Macaulay of dimension $\dim (A) \geq 4$,

  4. condition (2) of Lemma 52.27.3 holds by induction hypothesis as $\dim ((A'_{f_ r})_\mathfrak p) \geq 4$ 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 prime ideal $\mathfrak q \subset B$ and $B_\mathfrak q$ is regular.

This finishes the proof. $\square$

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