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*}

Lemma 15.78.11. Let $A$ be a ring. Let $E \to D \to F \to E[1]$ be a distinguished triangle of $D(\mathbf{N}, A)$. Let $(E_ n)$, resp. $(D_ n)$, resp. $(F_ n)$ be the system of objects of $D(A)$ associated to $E$, resp. $D$, resp. $F$. Then for every $K \in D(A)$ there is a canonical distinguished triangle

\[ R\mathop{\mathrm{lim}}\nolimits (K \otimes ^\mathbf {L}_ A E_ n) \to R\mathop{\mathrm{lim}}\nolimits (K \otimes ^\mathbf {L}_ A D_ n) \to R\mathop{\mathrm{lim}}\nolimits (K \otimes ^\mathbf {L}_ A F_ n) \to R\mathop{\mathrm{lim}}\nolimits (K \otimes ^\mathbf {L}_ A E_ n)[1] \]

in $D(A)$ with notation as in Remark 15.78.10.

Proof. This is clear from the construction in Remark 15.78.10 and the fact that $\Delta : D(A) \to D(\mathbf{N}, A)$, $- \otimes ^\mathbf {L} -$, and $R\mathop{\mathrm{lim}}\nolimits $ are exact functors of triangulated categories. $\square$


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