Lemma 20.36.4. Let A be a ring. Let f \in A. Let X be a topological space. Let
\ldots \to \mathcal{F}_3 \to \mathcal{F}_2 \to \mathcal{F}_1
be an inverse system of sheaves of A-modules. Let p \geq 0. Assume
either H^{p + 1}(X, \mathcal{F}_1) is an A-module of finite length or A is Noetherian and H^{p + 1}(X, \mathcal{F}_1) is a finite A-module,
X, f, (\mathcal{F}_ n) satisfy condition (1) of Lemma 20.36.1.
Then the inverse system M_ n = H^ p(X, \mathcal{F}_ n) satisfies the Mittag-Leffler condition.
Proof.
Set I = (f). We will use the criterion of Lemma 20.35.1. Observe that f^ n : \mathcal{F}_1 \to I^ n\mathcal{F}_{n + 1} is an isomorphism for all n \geq 0. Thus it suffices to show that
\bigoplus \nolimits _{n \geq 1} H^{p + 1}(X, \mathcal{F}_1) \cdot f^{n + 1}
is a graded S = \bigoplus _{n \geq 0} A/(f) \cdot f^ n-module satisfying the ascending chain condition. If A is not Noetherian, then H^{p + 1}(X, \mathcal{F}_1) has finite length and the result holds. If A is Noetherian, then S is a Noetherian ring and the result holds as the module is finite over S by the assumed finiteness of H^{p + 1}(X, \mathcal{F}_1). Some details omitted.
\square
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