The Stacks project

Lemma 22.6.2. Let $(A, \text{d})$ be a differential graded algebra. Suppose that

\[ \xymatrix{ K_1 \ar[r]_{f_1} \ar[d]_ a & L_1 \ar[d]^ b \\ K_2 \ar[r]^{f_2} & L_2 } \]

is a diagram of homomorphisms of differential graded $A$-modules which is commutative up to homotopy. Then there exists a morphism $c : C(f_1) \to C(f_2)$ which gives rise to a morphism of triangles

\[ (a, b, c) : (K_1, L_1, C(f_1), f_1, i_1, p_1) \to (K_1, L_1, C(f_1), f_2, i_2, p_2) \]

in $K(\text{Mod}_{(A, \text{d})})$.

Proof. Let $h : K_1 \to L_2$ be a homotopy between $f_2 \circ a$ and $b \circ f_1$. Define $c$ by the matrix

\[ c = \left( \begin{matrix} b & h \\ 0 & a \end{matrix} \right) : L_1 \oplus K_1 \to L_2 \oplus K_2 \]

A matrix computation show that $c$ is a morphism of differential graded modules. It is trivial that $c \circ i_1 = i_2 \circ b$, and it is trivial also to check that $p_2 \circ c = a \circ p_1$. $\square$

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