The Stacks project

Lemma 22.7.6. Let $(A, \text{d})$ be a differential graded algebra. Let $0 \to K_ i \to L_ i \to M_ i \to 0$, $i = 1, 2, 3$ be admissible short exact sequence of differential graded $A$-modules. Let $b : L_1 \to L_2$ and $b' : L_2 \to L_3$ be homomorphisms of differential graded modules such that

\[ \vcenter { \xymatrix{ K_1 \ar[d]_0 \ar[r] & L_1 \ar[r] \ar[d]_ b & M_1 \ar[d]_0 \\ K_2 \ar[r] & L_2 \ar[r] & M_2 } } \quad \text{and}\quad \vcenter { \xymatrix{ K_2 \ar[d]^0 \ar[r] & L_2 \ar[r] \ar[d]^{b'} & M_2 \ar[d]^0 \\ K_3 \ar[r] & L_3 \ar[r] & M_3 } } \]

commute up to homotopy. Then $b' \circ b$ is homotopic to $0$.

Proof. By Lemma 22.7.3 we can replace $b$ and $b'$ by homotopic maps such that the right square of the left diagram commutes and the left square of the right diagram commutes. In other words, we have $\mathop{\mathrm{Im}}(b) \subset \mathop{\mathrm{Im}}(K_2 \to L_2)$ and $\mathop{\mathrm{Ker}}((b')^ n) \supset \mathop{\mathrm{Im}}(K_2 \to L_2)$. Then $b \circ b' = 0$ as a map of modules. $\square$

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