The Stacks project

Lemma 22.7.2. 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. Let $s : M \to L$ and $\pi : L \to K$ be splittings such that $\mathop{\mathrm{Ker}}(\pi ) = \mathop{\mathrm{Im}}(s)$. Then we obtain a morphism

\[ \delta = \pi \circ \text{d}_ L \circ s : M \to K[1] \]

of $\text{Mod}_{(A, \text{d})}$ which induces the boundary maps in the long exact sequence of cohomology (

Proof. The map $\pi \circ \text{d}_ L \circ s$ is compatible with the $A$-module structure and the gradings by construction. It is compatible with differentials by Homology, Lemmas 12.14.10. Let $R$ be the ring that $A$ is a differential graded algebra over. The equality of maps is a statement about $R$-modules. Hence this follows from Homology, Lemmas 12.14.10 and 12.14.11. $\square$

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