The Stacks project

Remark 52.20.7. Let $(A, \mathfrak m)$ be a Noetherian local ring which has a dualizing complex and is complete with respect to $f \in \mathfrak m$. Let $(\mathcal{F}_ n)$ be an object of $\textit{Coh}(U, f\mathcal{O}_ U)$ where $U$ is the punctured spectrum of $A$. Set $Y = V(f) \subset X = \mathop{\mathrm{Spec}}(A)$. If for $y \in U \cap V(f)$ closed in $U$, i.e., with $\dim (\overline{\{ y\} }) = 1$, we assume the $\mathcal{O}_{X, y}^\wedge $-module $\mathcal{F}_ y^\wedge $ satisfies the following two conditions

  1. $\mathcal{F}_ y^\wedge [1/f]$ is $(S_2)$ as a $\mathcal{O}_{X, y}^\wedge [1/f]$-module, and

  2. for $\mathfrak p \in \text{Ass}(\mathcal{F}_ y^\wedge [1/f])$ we have $\dim (\mathcal{O}_{X, y}^\wedge /\mathfrak p) \geq 3$.

Then $(\mathcal{F}_ n)$ is the completion of a coherent module on $U$. This follows from Lemmas 52.20.4 and 52.20.5.

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