Lemma 22.9.2 (09KC)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 22: Differential Graded Algebra
Section 22.9: Cones and distinguished triangles
Lemma 22.9.2 (cite)

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Lemma 22.9.2. Let $(A, \text{d})$ be a differential graded algebra. Let $\alpha : K \to L$ and $\beta : L \to M$ define an admissible short exact sequence

\[ 0 \to K \to L \to M \to 0 \]

of differential graded $A$-modules. Let $(K, L, M, \alpha , \beta , \delta )$ be the associated triangle. Then the triangles

\[ (M[-1], K, L, \delta [-1], \alpha , \beta ) \quad \text{and}\quad (M[-1], K, C(\delta [-1]), \delta [-1], i, p) \]

are isomorphic.

Proof. Using a choice of splittings we write $L = K \oplus M$ and we identify $\alpha $ and $\beta $ with the natural inclusion and projection maps. By construction of $\delta $ we have

\[ d_ B = \left( \begin{matrix} d_ K & \delta \\ 0 & d_ M \end{matrix} \right) \]

On the other hand the cone of $\delta [-1] : M[-1] \to K$ is given as $C(\delta [-1]) = K \oplus M$ with differential identical with the matrix above! Whence the lemma. $\square$

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