Proposition 21.51.2. Let $\mathcal{C}$ be a site. Let $\mathcal{B} \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ such that every $U \in \mathcal{B}$ is weakly contractible and every object of $\mathcal{C}$ has a covering by elements of $\mathcal{B}$. Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}$. Then

A complex $\mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3$ of $\mathcal{O}$-modules is exact, if and only if $\mathcal{F}_1(U) \to \mathcal{F}_2(U) \to \mathcal{F}_3(U)$ is exact for all $U \in \mathcal{B}$.

Every object $K$ of $D(\mathcal{O})$ is a derived limit of its canonical truncations: $K = R\mathop{\mathrm{lim}}\nolimits \tau _{\geq -n} K$.

Given an inverse system $\ldots \to \mathcal{F}_3 \to \mathcal{F}_2 \to \mathcal{F}_1$ with surjective transition maps, the projection $\mathop{\mathrm{lim}}\nolimits \mathcal{F}_ n \to \mathcal{F}_1$ is surjective.

Products are exact on $\textit{Mod}(\mathcal{O})$.

Products on $D(\mathcal{O})$ can be computed by taking products of any representative complexes.

If $(\mathcal{F}_ n)$ is an inverse system of $\mathcal{O}$-modules, then $R^ p\mathop{\mathrm{lim}}\nolimits \mathcal{F}_ n = 0$ for all $p > 1$ and

\[ R^1\mathop{\mathrm{lim}}\nolimits \mathcal{F}_ n = \mathop{\mathrm{Coker}}(\prod \mathcal{F}_ n \to \prod \mathcal{F}_ n) \]where the map is $(x_ n) \mapsto (x_ n - f(x_{n + 1}))$.

If $(K_ n)$ is an inverse system of objects of $D(\mathcal{O})$, then there are short exact sequences

\[ 0 \to R^1\mathop{\mathrm{lim}}\nolimits H^{p - 1}(K_ n) \to H^ p(R\mathop{\mathrm{lim}}\nolimits K_ n) \to \mathop{\mathrm{lim}}\nolimits H^ p(K_ n) \to 0 \]

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