Lemma 20.38.2. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $(\mathcal{F}_ n^\bullet )$ be an inverse system of complexes of $\mathcal{O}_ X$-modules. Let $m \in \mathbf{Z}$. Assume there exist a set $\mathcal{B}$ of open subsets of $X$ and an integer $n_0$ such that

every open in $X$ has a covering whose members are elements of $\mathcal{B}$,

for every $U \in \mathcal{B}$

the systems of abelian groups $\mathcal{F}_ n^{m - 2}(U)$ and $\mathcal{F}_ n^{m - 1}(U)$ have vanishing $R^1\mathop{\mathrm{lim}}\nolimits $ (for example these have the Mittag-Leffler condition),

the system of abelian groups $H^{m - 1}(\mathcal{F}_ n^\bullet (U))$ has vanishing $R^1\mathop{\mathrm{lim}}\nolimits $ (for example it has the Mittag-Leffler condition), and

we have $H^ m(\mathcal{F}_ n^\bullet (U)) = H^ m(\mathcal{F}_{n_0}^\bullet (U))$ for all $n \geq n_0$.

Then the maps $H^ m(\mathcal{F}^\bullet ) \to \mathop{\mathrm{lim}}\nolimits H^ m(\mathcal{F}_ n^\bullet ) \to H^ m(\mathcal{F}_{n_0}^\bullet )$ are isomorphisms of sheaves where $\mathcal{F}^\bullet = \mathop{\mathrm{lim}}\nolimits \mathcal{F}_ n^\bullet $ is the termwise inverse limit.

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