The Stacks project

Lemma 22.7.3. Let $(A, \text{d})$ be a differential graded algebra. Let

\[ \xymatrix{ K \ar[r]_ f \ar[d]_ a & L \ar[d]^ b \\ M \ar[r]^ g & N } \]

be a diagram of homomorphisms of differential graded $A$-modules commuting up to homotopy.

  1. If $f$ is an admissible monomorphism, then $b$ is homotopic to a homomorphism which makes the diagram commute.

  2. If $g$ is an admissible epimorphism, then $a$ is homotopic to a morphism which makes the diagram commute.

Proof. Let $h : K \to N$ be a homotopy between $bf$ and $ga$, i.e., $bf - ga = \text{d}h + h\text{d}$. Suppose that $\pi : L \to K$ is a graded $A$-module map left inverse to $f$. Take $b' = b - \text{d}h\pi - h\pi \text{d}$. Suppose $s : N \to M$ is a graded $A$-module map right inverse to $g$. Take $a' = a + \text{d}sh + sh\text{d}$. Computations omitted. $\square$

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