The Stacks project

Lemma 21.24.2. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(\mathcal{F}_ n^\bullet )$ be an inverse system of complexes of $\mathcal{O}$-modules. Let $m \in \mathbf{Z}$. Suppose given $\mathcal{B} \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and an integer $n_0$ such that

  1. every object of $\mathcal{C}$ has a covering whose members are elements of $\mathcal{B}$,

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

    1. 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 property),

    2. 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 property), and

    3. 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.

Proof. Let $U \in \mathcal{B}$. Note that $H^ m(\mathcal{F}^\bullet (U))$ is the cohomology of

\[ \mathop{\mathrm{lim}}\nolimits _ n \mathcal{F}_ n^{m - 2}(U) \to \mathop{\mathrm{lim}}\nolimits _ n \mathcal{F}_ n^{m - 1}(U) \to \mathop{\mathrm{lim}}\nolimits _ n \mathcal{F}_ n^ m(U) \to \mathop{\mathrm{lim}}\nolimits _ n \mathcal{F}_ n^{m + 1}(U) \]

in the third spot from the left. By assumptions (2)(a) and (2)(b) we may apply More on Algebra, Lemma 15.86.3 to conclude that

\[ H^ m(\mathcal{F}^\bullet (U)) = \mathop{\mathrm{lim}}\nolimits H^ m(\mathcal{F}_ n^\bullet (U)) \]

By assumption (2)(c) we conclude

\[ H^ m(\mathcal{F}^\bullet (U)) = H^ m(\mathcal{F}_ n^\bullet (U)) \]

for all $n \geq n_0$. By assumption (1) we conclude that the sheafification of $U \mapsto H^ m(\mathcal{F}^\bullet (U))$ is equal to the sheafification of $U \mapsto H^ m(\mathcal{F}_ n^\bullet (U))$ for all $n \geq n_0$. Thus the inverse system of sheaves $H^ m(\mathcal{F}_ n^\bullet )$ is constant for $n \geq n_0$ with value $H^ m(\mathcal{F}^\bullet )$ which proves the lemma. $\square$

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