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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