The Stacks project

Lemma 52.27.1. Let $(A, \mathfrak m)$ be a Noetherian local ring and $f \in \mathfrak m$. Assume

  1. $A$ is $f$-adically complete,

  2. $f$ is a nonzerodivisor,

  3. $H^1_\mathfrak m(A/fA)$ and $H^2_\mathfrak m(A/fA)$ are finite $A$-modules, and

  4. $H^3_\mathfrak m(A/fA) = 0$1.

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

\[ \mathop{\mathrm{colim}}\nolimits _{U_0 \subset U' \subset U\text{ open}} \mathop{\mathrm{Pic}}\nolimits (U') \longrightarrow \mathop{\mathrm{Pic}}\nolimits (U_0) \]

is surjective.

Proof. Let $U_0 \subset U_ n \subset U$ be the $n$th infinitesimal neighbourhood of $U_0$. Observe that the ideal sheaf of $U_ n$ in $U_{n + 1}$ is isomorphic to $\mathcal{O}_{U_0}$ as $U_0 \subset U$ is the principal closed subscheme cut out by the nonzerodivisor $f$. Hence we have an exact sequence of abelian groups

\[ \mathop{\mathrm{Pic}}\nolimits (U_{n + 1}) \to \mathop{\mathrm{Pic}}\nolimits (U_ n) \to H^2(U_0, \mathcal{O}_{U_0}) = H^3_\mathfrak m(A/fA) = 0 \]

see More on Morphisms, Lemma 37.4.1. Thus every invertible $\mathcal{O}_{U_0}$-module is the restriction of an invertible coherent formal module, i.e., an invertible object of $\textit{Coh}(U, f\mathcal{O}_ U)$. We conclude by applying Lemma 52.24.2. $\square$

[1] Observe that (3) and (4) hold if $\text{depth}(A/fA) \geq 4$, or equivalently $\text{depth}(A) \geq 5$.

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