The Stacks project

Lemma 52.5.3. Let $(A, \mathfrak m)$ be a Noetherian local ring. Let $(M_ n)$ be an inverse system of finite $A$-modules. Let $M \to \mathop{\mathrm{lim}}\nolimits M_ n$ be a map where $M$ is a finite $A$-module such that for some $i$ the map $H^ i_\mathfrak m(M) \to \mathop{\mathrm{lim}}\nolimits H^ i_\mathfrak m(M_ n)$ is an isomorphism. Then the inverse system $H^ i_\mathfrak m(M_ n)$ is essentially constant with value $H^ i_\mathfrak m(M)$.

Proof. By Lemma 52.5.2 the inverse system $H^ i_\mathfrak m(M_ n)$ satisfies the Mittag-Leffler condition. Let $E_ n \subset H^ i_\mathfrak m(M_ n)$ be the image of $H^ i_\mathfrak m(M_{n'})$ for $n' \gg n$. Then $(E_ n)$ is an inverse system with surjective transition maps and $H^ i_\mathfrak m(M) = \mathop{\mathrm{lim}}\nolimits E_ n$. Since $H^ i_\mathfrak m(M)$ has the descending chain condition by Lemma 52.5.1 we find there can only be a finite number of nontrivial kernels of the surjections $H^ i_\mathfrak m(M) \to E_ n$. Thus $E_ n \to E_{n - 1}$ is an isomorphism for all $n \gg 0$ as desired. $\square$

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