The Stacks project

Lemma 52.5.2. Let $(A, \mathfrak m)$ be a Noetherian local ring.

  1. Let $(M_ n)$ be an inverse system of finite $A$-modules. Then the inverse system $H^ i_\mathfrak m(M_ n)$ satisfies the Mittag-Leffler condition for any $i$.

  2. Let $U = \mathop{\mathrm{Spec}}(A) \setminus \{ \mathfrak m\} $ be the punctured spectrum of $A$. Let $\mathcal{F}_ n$ be an inverse system of coherent $\mathcal{O}_ U$-modules. Then the inverse system $H^ i(U, \mathcal{F}_ n)$ satisfies the Mittag-Leffler condition for $i > 0$.

Proof. Follows immediately from Lemma 52.5.1. $\square$

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