Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{F}^\bullet $ be a complex of $\mathcal{O}$-modules. The category $\textit{Mod}(\mathcal{O})$ has enough injectives, hence we can use Derived Categories, Lemma 13.29.3 produce a diagram
in the category of complexes of $\mathcal{O}$-modules such that
the vertical arrows are quasi-isomorphisms,
$\mathcal{I}_ n^\bullet $ is a bounded below complex of injectives,
the arrows $\mathcal{I}_{n + 1}^\bullet \to \mathcal{I}_ n^\bullet $ are termwise split surjections.
The category of $\mathcal{O}$-modules has limits (they are computed on the level of presheaves), hence we can form the termwise limit $\mathcal{I}^\bullet = \mathop{\mathrm{lim}}\nolimits _ n \mathcal{I}_ n^\bullet $. By Derived Categories, Lemmas 13.31.4 and 13.31.8 this is a K-injective complex. In general the canonical map
may not be a quasi-isomorphism. In the following lemma we describe some conditions under which it is.
Lemma 21.24.1. In the situation described above. Denote $\mathcal{H}^ m = H^ m(\mathcal{F}^\bullet )$ the $m$th cohomology sheaf. Let $\mathcal{B} \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ be a subset. Let $d \in \mathbf{N}$. Assume
every object of $\mathcal{C}$ has a covering whose members are elements of $\mathcal{B}$,
for every $U \in \mathcal{B}$ we have $H^ p(U, \mathcal{H}^ q) = 0$ for $p > d$ and $q < 0$1.
Then (21.24.0.1) is a quasi-isomorphism.
Proof. By Derived Categories, Lemma 13.34.5 it suffices to show that the map $\mathcal{F}^\bullet \to R\mathop{\mathrm{lim}}\nolimits \tau _{\geq -n} \mathcal{F}^\bullet $ is an isomorphism. This follows from Lemma 21.23.10. $\square$
Here is a technical lemma about cohomology sheaves of termwise limits of inverse systems of complexes of modules. We should avoid using this lemma as much as possible and instead use arguments with derived inverse limits.
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
every object of $\mathcal{C}$ 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 property),
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
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
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
By assumption (2)(c) we conclude
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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