The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

Remark 15.77.9. Let $(K_ n)$ be an inverse system of objects of $D(\textit{Ab})$. Let $K = R\mathop{\mathrm{lim}}\nolimits K_ n$ be a derived limit of this system (see Derived Categories, Section 13.32). Such a derived limit exists because $D(\textit{Ab})$ has countable products (Derived Categories, Lemma 13.32.2). By Lemma 15.77.8 we can also lift $(K_ n)$ to an object $M$ of $D(\mathbf{N})$. Then $K \cong R\mathop{\mathrm{lim}}\nolimits M$ where $R\mathop{\mathrm{lim}}\nolimits $ is the functor (15.77.1.1) because $R\mathop{\mathrm{lim}}\nolimits M$ is also a derived limit of the system $(K_ n)$ by Lemma 15.77.6. Thus, although there may be many isomorphism classes of lifts $M$ of the system $(K_ n)$, the isomorphism type of $R\mathop{\mathrm{lim}}\nolimits M$ is independent of the choice because it is isomorphic to the derived limit $K = R\mathop{\mathrm{lim}}\nolimits K_ n$ of the system. Thus we may apply results on $R\mathop{\mathrm{lim}}\nolimits $ proved in this section to derived limits. For example, for every $p \in \mathbf{Z}$ there is a canonical short exact sequence

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

because we may apply Lemma 15.77.6 to $M$. This can also been seen directly, without invoking the existence of $M$, by applying the argument of the proof of Lemma 15.77.6 to the (defining) distinguished triangle $K \to \prod K_ n \to \prod K_ n \to K[1]$.


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