Lemma 10.134.14. Let $R$ be a ring. Let $A_1 \to A_0$, and $B_1 \to B_0$ be two term complexes. Suppose that there exist morphisms of complexes $\varphi : A_\bullet \to B_\bullet $ and $\psi : B_\bullet \to A_\bullet $ such that $\varphi \circ \psi $ and $\psi \circ \varphi $ are homotopic to the identity maps. Then $A_1 \oplus B_0 \cong B_1 \oplus A_0$ as $R$-modules.
Proof. Choose a map $h : A_0 \to A_1$ such that
Similarly, choose a map $h' : B_0 \to B_1$ such that
A trivial computation shows that
This shows that both matrices on the right hand side are invertible and proves the lemma. $\square$
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