The Stacks project

Lemma 30.23.1. If $X = \mathop{\mathrm{Spec}}(A)$ is the spectrum of a Noetherian ring and $\mathcal{I}$ is the quasi-coherent sheaf of ideals associated to the ideal $I \subset A$, then $\textit{Coh}(X, \mathcal{I})$ is equivalent to the category of finite $A^\wedge $-modules where $A^\wedge $ is the completion of $A$ with respect to $I$.

Proof. Let $\text{Mod}^{fg}_{A, I}$ be the category of inverse systems $(M_ n)$ of finite $A$-modules satisfying: (1) $M_ n$ is annihilated by $I^ n$ and (2) $M_{n + 1}/I^ nM_{n + 1} = M_ n$. By the correspondence between coherent sheaves on $X$ and finite $A$-modules (Lemma 30.9.1) it suffices to show $\text{Mod}^{fg}_{A, I}$ is equivalent to the category of finite $A^\wedge $-modules. To see this it suffices to prove that given an object $(M_ n)$ of $\text{Mod}^{fg}_{A, I}$ the module

\[ M = \mathop{\mathrm{lim}}\nolimits M_ n \]

is a finite $A^\wedge $-module and that $M/I^ nM = M_ n$. As the transition maps are surjective, we see that $M \to M_1$ is surjective. Pick $x_1, \ldots , x_ t \in M$ which map to generators of $M_1$. This induces a map of systems $(A/I^ n)^{\oplus t} \to M_ n$. By Nakayama's lemma (Algebra, Lemma 10.20.1) these maps are surjective. Let $K_ n \subset (A/I^ n)^{\oplus t}$ be the kernel. Property (2) implies that $K_{n + 1} \to K_ n$ is surjective, in particular the system $(K_ n)$ satisfies the Mittag-Leffler condition. By Homology, Lemma 12.31.3 we obtain an exact sequence $0 \to K \to (A^\wedge )^{\oplus t} \to M \to 0$ with $K = \mathop{\mathrm{lim}}\nolimits K_ n$. Hence $M$ is a finite $A^\wedge $-module. As $K \to K_ n$ is surjective it follows that

\[ M/I^ nM = \mathop{\mathrm{Coker}}(K \to (A/I^ n)^{\oplus t}) = (A/I^ n)^{\oplus t}/K_ n = M_ n \]

as desired. $\square$

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