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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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