The Stacks project

Lemma 13.9.5. Let $\mathcal{A}$ be an additive category. Let

\[ \xymatrix{ A^\bullet \ar[r]_ f \ar[d]_ a & B^\bullet \ar[d]^ b \\ C^\bullet \ar[r]^ g & D^\bullet } \]

be a diagram of morphisms of complexes commuting up to homotopy. If $f$ is a termwise split injection, then $b$ is homotopic to a morphism which makes the diagram commute. If $g$ is a split surjection, then $a$ is homotopic to a morphism which makes the diagram commute.

Proof. Let $h^ n : A^ n \to D^{n - 1}$ be a collection of morphisms such that $bf - ga = dh + hd$. Suppose that $\pi ^ n : B^ n \to A^ n$ are morphisms splitting the morphisms $f^ n$. Take $b' = b - dh\pi - h\pi d$. Suppose $s^ n : D^ n \to C^ n$ are morphisms splitting the morphisms $g^ n : C^ n \to D^ n$. Take $a' = a + dsh + shd$. Computations omitted. $\square$

Comments (1)

Comment #292 by arp on

Nitpicky typos: I think you should define . Also, in the sentence right after the proof of this lemma, "an morphism of complexes" should be replaced with "a morphism of complexes," ha...

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