The Stacks project

Lemma 52.27.4. Let $(A, \mathfrak m)$ be a Noetherian local ring of depth $\geq 2$. Let $A^\wedge $ be its completion. Let $U$, resp. $U^\wedge $ be the punctured spectrum of $A$, resp. $A^\wedge $. Then $\mathop{\mathrm{Pic}}\nolimits (U) \to \mathop{\mathrm{Pic}}\nolimits (U^\wedge )$ is injective.

Proof. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ U$-module with pullback $\mathcal{L}^\wedge $ on $U^\wedge $. We have $H^0(U, \mathcal{O}_ U) = A$ by our assumption on depth and Dualizing Complexes, Lemma 47.11.1 and Local Cohomology, Lemma 51.8.2. Thus $\mathcal{L}$ is trivial if and only if $M = H^0(U, \mathcal{L})$ is isomorphic to $A$ as an $A$-module. (Details omitted.) Since $A \to A^\wedge $ is flat we have $M \otimes _ A A^\wedge = \Gamma (U^\wedge , \mathcal{L}^\wedge )$ by flat base change, see Cohomology of Schemes, Lemma 30.5.2. Finally, it is easy to see that $M \cong A$ if and only if $M \otimes _ A A^\wedge \cong A^\wedge $. $\square$

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