The Stacks project

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

  1. 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,

  2. $X$, $f$, $(\mathcal{F}_ n)$ 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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