The Stacks project

Lemma 52.16.2. In Situation 52.16.1. Consider an inverse system $(M_ n)$ of $A$-modules such that

  1. $M_ n$ is a finite $A$-module,

  2. $M_ n$ is annihilated by $I^ n$,

  3. the kernel and cokernel of $M_{n + 1}/I^ nM_{n + 1} \to M_ n$ are $\mathfrak a$-power torsion.

Then $(\widetilde{M}_ n|_ U)$ is in $\textit{Coh}(U, I\mathcal{O}_ U)$. Conversely, every object of $\textit{Coh}(U, I\mathcal{O}_ U)$ arises in this manner.

Proof. We omit the verification that $(\widetilde{M}_ n|_ U)$ is in $\textit{Coh}(U, I\mathcal{O}_ U)$. Let $(\mathcal{F}_ n)$ be an object of $\textit{Coh}(U, I\mathcal{O}_ U)$. By Local Cohomology, Lemma 51.8.2 we see that $\mathcal{F}_ n = \widetilde{M_ n}$ for some finite $A/I^ n$-module $M_ n$. After dividing $M_ n$ by $H^0_\mathfrak a(M_ n)$ we may assume $M_ n \subset H^0(U, \mathcal{F}_ n)$, see Dualizing Complexes, Lemma 47.11.6 and the already referenced lemma. After replacing inductively $M_{n + 1}$ by the inverse image of $M_ n$ under the map $M_{n + 1} \to H^0(U, \mathcal{F}_{n + 1}) \to H^0(U, \mathcal{F}_ n)$, we may assume $M_{n + 1}$ maps into $M_ n$. This gives a inverse system $(M_ n)$ satisfying (1) and (2) such that $\mathcal{F}_ n = \widetilde{M_ n}$. To see that (3) holds, use that $M_{n + 1}/I^ nM_{n + 1} \to M_ n$ is a map of finite $A$-modules which induces an isomorphism after applying $\widetilde{\ }$ and restriction to $U$ (here we use the first referenced lemma one more time). $\square$

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