The Stacks project

Lemma 20.36.5. 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 there is an $m \geq 1$ such that the image of $H^{p + 1}(X, \mathcal{F}_ m) \to H^{p + 1}(X, \mathcal{F}_1)$ is an $A$-module of finite length or $A$ is Noetherian and the intersection of the images of $H^{p + 1}(X, \mathcal{F}_ m) \to H^{p + 1}(X, \mathcal{F}_1)$ 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 20.35.2 involving the modules $N_ n$. For $m \geq n$ we have $I^ n\mathcal{F}_{m + 1} = \mathcal{F}_{m + 1 - n}$. Thus we see that

\[ N_ n = \bigcap \nolimits _{m \geq 1} \mathop{\mathrm{Im}}\left( H^{p + 1}(X, \mathcal{F}_ m) \to H^{p + 1}(X, \mathcal{F}_1) \right) \]

is independent of $n$ and $\bigoplus N_ n = \bigoplus N_1 \cdot f^{n + 1}$. Thus we conclude exactly as in the proof of Lemma 20.36.4. $\square$

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