Lemma 52.2.2. Let $I$ be an ideal of a ring $A$. Let $X$ be a scheme over $\mathop{\mathrm{Spec}}(A)$. Let

$\ldots \to \mathcal{F}_3 \to \mathcal{F}_2 \to \mathcal{F}_1$

be an inverse system of $\mathcal{O}_ X$-modules such that $\mathcal{F}_ n = \mathcal{F}_{n + 1}/I^ n\mathcal{F}_{n + 1}$. Given $n$ define

$H^1_ n = \bigcap \nolimits _{m \geq n} \mathop{\mathrm{Im}}\left( H^1(X, I^ n\mathcal{F}_{m + 1}) \to H^1(X, I^ n\mathcal{F}_{n + 1}) \right)$

If $\bigoplus H^1_ n$ satisfies the ascending chain condition as a graded $\bigoplus _{n \geq 0} I^ n/I^{n + 1}$-module, then the inverse system $M_ n = \Gamma (X, \mathcal{F}_ n)$ satisfies the Mittag-Leffler condition.

Proof. This is a special case of the more general Cohomology, Lemma 20.35.2. $\square$

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