Lemma 52.3.5. Let $A$ be a ring. Let $f \in A$. Let $X$ be a scheme over $\mathop{\mathrm{Spec}}(A)$. Let

be an inverse system of $\mathcal{O}_ X$-modules. Assume

either there is an $m \geq 1$ such that the image of $H^1(X, \mathcal{F}_ m) \to H^1(X, \mathcal{F}_1)$ is an $A$-module of finite length or $A$ is Noetherian and the intersection of the images of $H^1(X, \mathcal{F}_ m) \to H^1(X, \mathcal{F}_1)$ is a finite $A$-module,

the equivalent conditions of Lemma 52.3.1 hold.

Then the inverse system $M_ n = \Gamma (X, \mathcal{F}_ n)$ satisfies the Mittag-Leffler condition.

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