The Stacks project

Lemma 52.26.1. Let $(A, \mathfrak m)$ be a Noetherian local ring. Let $f \in \mathfrak m$ be a nonzerodivisor and assume that $\text{depth}(A/fA) \geq 2$, or equivalently $\text{depth}(A) \geq 3$. Let $U$, resp. $U_0$ be the punctured spectrum of $A$, resp. $A/fA$. The map

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

is injective on torsion.

Proof. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ U$-module. Observe that $\mathcal{L}$ maps to $0$ in $\mathop{\mathrm{Pic}}\nolimits (U_0)$ if and only if we can extend $\mathcal{L}$ to an invertible coherent triple $(\mathcal{L}, \mathcal{L}_0, \lambda )$ as in Section 52.25. By Proposition 52.25.4 the function

\[ n \longmapsto \chi ((\mathcal{L}, \mathcal{L}_0, \lambda )^{\otimes n}) \]

is a polynomial. By Lemma 52.25.5 the value of this polynomial is zero if and only if $\mathcal{L}^{\otimes n}$ is trivial. Thus if $\mathcal{L}$ is torsion, then this polynomial has infinitely many zeros, hence is identically zero, hence $\mathcal{L}$ is trivial. $\square$

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