The Stacks project

Remark 52.26.3. In SGA2 we find the following result. Let $(A, \mathfrak m)$ be a Noetherian local ring. Let $f \in \mathfrak m$. Assume $A$ is a quotient of a regular ring, the element $f$ is a nonzerodivisor, and

  1. if $\mathfrak p \subset A$ is a prime ideal with $\dim (A/\mathfrak p) = 1$, then $\text{depth}(A_\mathfrak p) \geq 2$, and

  2. $\text{depth}(A/fA) \geq 3$, or equivalently $\text{depth}(A) \geq 4$.

Let $U$, resp. $U_0$ be the punctured spectrum of $A$, resp. $A/fA$. Then the map

\[ \mathop{\mathrm{Pic}}\nolimits (U) \to \mathop{\mathrm{Pic}}\nolimits (U_0) \]

is injective. This is [Exposee XI, Lemma 3.16, SGA2]1. This result from SGA2 follows from Proposition 52.26.2 because

  1. a quotient of a regular ring has a dualizing complex (see Dualizing Complexes, Lemma 47.21.3 and Proposition 47.15.11), and

  2. if $\text{depth}(A) \geq 4$ then $\text{depth}(A_\mathfrak p) \geq 2$ for all primes $\mathfrak p$ with $\dim (A/\mathfrak p) = 2$, see Algebra, Lemma 10.72.10.

[1] Condition (a) follows from condition (b), see Algebra, Lemma 10.72.10.

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