The Stacks project

Lemma 52.16.6. In Situation 52.16.1 let $(\mathcal{F}_ n)$ be an object of $\textit{Coh}(U, I\mathcal{O}_ U)$. Let $A', I', \mathfrak a'$ be the $I$-adic completions of $A, I, \mathfrak a$. Set $X' = \mathop{\mathrm{Spec}}(A')$ and $U' = X' \setminus V(\mathfrak a')$. The following are equivalent

  1. $(\mathcal{F}_ n)$ extends to $X$, and

  2. the pullback of $(\mathcal{F}_ n)$ to $U'$ is the completion of a coherent $\mathcal{O}_{U'}$-module.

Proof. Recall that $A \to A'$ is a flat ring map which induces an isomorphism $A/I \to A'/I'$. See Algebra, Lemmas 10.97.2 and 10.97.4. Thus $X' \to X$ is a flat morphism inducing an isomorphism $Y' \to Y$. Thus $U' \to U$ is a flat morphism which induces an isomorphism $U' \cap Y' \to U \cap Y$. This implies that in the commutative diagram

\[ \xymatrix{ \textit{Coh}(X', I\mathcal{O}_{X'}) \ar[r] & \textit{Coh}(U', I\mathcal{O}_{U'}) \\ \textit{Coh}(X, I\mathcal{O}_ X) \ar[u] \ar[r] & \textit{Coh}(U, I\mathcal{O}_ U) \ar[u] } \]

the vertical functors are equivalences. See Cohomology of Schemes, Lemma 30.23.10. The lemma follows formally from this and the results of Lemma 52.16.3. $\square$

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