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*}

22.8 Distinguished triangles

The following lemma produces our distinguished triangles.

Lemma 22.8.1. Let $(A, \text{d})$ be a differential graded algebra. Let $0 \to K \to L \to M \to 0$ be an admissible short exact sequence of differential graded $A$-modules. The triangle

22.8.1.1
\begin{equation} \label{dga-equation-triangle-associated-to-admissible-ses} K \to L \to M \xrightarrow {\delta } K[1] \end{equation}

with $\delta $ as in Lemma 22.7.2 is, up to canonical isomorphism in $K(\text{Mod}_{(A, \text{d})})$, independent of the choices made in Lemma 22.7.2.

Proof. Namely, let $(s', \pi ')$ be a second choice of splittings as in Lemma 22.7.2. Then we claim that $\delta $ and $\delta '$ are homotopic. Namely, write $s' = s + \alpha \circ h$ and $\pi ' = \pi + g \circ \beta $ for some unique homomorphisms of $A$-modules $h : M \to K$ and $g : M \to K$ of degree $-1$. Then $g = -h$ and $g$ is a homotopy between $\delta $ and $\delta '$. The computations are done in the proof of Homology, Lemma 12.13.12. $\square$

Definition 22.8.2. Let $(A, \text{d})$ be a differential graded algebra.

  1. If $0 \to K \to L \to M \to 0$ is an admissible short exact sequence of differential graded $A$-modules, then the triangle associated to $0 \to K \to L \to M \to 0$ is the triangle (22.8.1.1) of $K(\text{Mod}_{(A, \text{d})})$.

  2. A triangle of $K(\text{Mod}_{(A, \text{d})})$ is called a distinguished triangle if it is isomorphic to a triangle associated to an admissible short exact sequence of differential graded $A$-modules.


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