The Stacks project

Lemma 52.16.11. In Situation 52.16.1 let $(\mathcal{F}_ n)$ be an object of $\textit{Coh}(U, I\mathcal{O}_ U)$. Assume

  1. $I = (f)$ is a principal ideal for a nonzerodivisor $f \in \mathfrak a$,

  2. $\mathcal{F}_ n$ is a finite locally free $\mathcal{O}_ U/f^ n\mathcal{O}_ U$-module,

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

Then $(\mathcal{F}_ n)$ extends canonically to $X$. In particular, if $A$ is complete, then $(\mathcal{F}_ n)$ is the completion of a coherent $\mathcal{O}_ U$-module.

Proof. We will prove this by verifying hypotheses (a), (b), and (c) of Lemma 52.16.10.

Since $\mathcal{F}_ n$ is locally free over $\mathcal{O}_ U/f^ n\mathcal{O}_ U$ we see that we have short exact sequences $0 \to \mathcal{F}_ n \to \mathcal{F}_{n + 1} \to \mathcal{F}_1 \to 0$ for all $n$. Thus condition (b) holds by Cohomology, Lemma 20.36.2.

As $f$ is a nonzerodivisor we obtain short exact sequences

\[ 0 \to A/f^ nA \xrightarrow {f} A/f^{n + 1}A \to A/fA \to 0 \]

and we have corresponding short exact sequences $0 \to \mathcal{F}_ n \to \mathcal{F}_{n + 1} \to \mathcal{F}_1 \to 0$. We will use Local Cohomology, Lemma 51.8.2 without further mention. Our assumptions imply that $H^0(U, \mathcal{O}_ U/f\mathcal{O}_ U)$ and $H^1(U, \mathcal{O}_ U/f\mathcal{O}_ U)$ are finite $A$-modules. Hence the same thing is true for $\mathcal{F}_1$, see Local Cohomology, Lemma 51.12.2. Using induction and the short exact sequences we find that $H^0(U, \mathcal{F}_ n)$ are finite $A$-modules for all $n$. In this way we see hypothesis (c) is satisfied.

Finally, as $H^1(U, \mathcal{F}_1)$ is a finite $A$-module we can apply Cohomology, Lemma 20.36.4 to see hypothesis (a) holds. $\square$

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