Lemma 22.8.1 (09K6)—The Stacks project

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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 12.14.12. $\square$

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