Proposition 6

Let $(C, d)$ be a complex. The following conditions are equivalent: (i) there exists a homotopy equivalence from $(C, d)$ to $(\mathrm{H}(C), 0)$; (ii) there exists an $A$-endomorphism $s$ of $C$, graded of degree 1, such that $d = d \circ s \circ d$; (iii) $B(C)$ and $Z(C)$ are direct summands of $C$; (iv) $(C, d)$ is a direct sum of subcomplexes which are either of length 0, or of length 1 and have zero homology.

(i) $\Rightarrow$ (ii): let $\varphi : C \to \mathrm{H}(C)$ be a homotopy equivalence; then there exists a morphism of complexes $\psi : \mathrm{H}(C) \to C$ and an endomorphism $s$ of $C$, graded of degree 1, such that $\psi \circ \varphi = 1_C - s \circ d - d \circ s$. We have $d \circ \psi = \psi \circ 0 = 0$, hence

whence (ii).

$(ii) \Rightarrow (iii)$: let $s$ be as in (ii). Then $d \circ (1_C-s\circ d)=0$, so $1_C-s\circ d$ is a projector of $C$ onto $Z(C)$, and $(d\circ s)\circ d=d$, so $d\circ s$ is a projector of $C$ onto $B(C)$.

$(iii) \Rightarrow (iv)$: for each $n\in\mathbf{Z}$, let $Z_n=Z_n(C)$, $B_n=B_n(C)$ and choose submodules $K_n$ and $B'_n$ of $C_n$ such that $C_n=B'_n\oplus Z_n$, $Z_n=K_n\oplus B_n$. Then

are subcomplexes of $(C,d)$; one has

; each $E_{(n)}$ is either zero or of length $0$, each $F_{(n)}$ is either zero or of length $1$ and of zero homology, whence (iv).

$(iv) \Rightarrow (i)$: it suffices to remark that (i) is satisfied when $C$ is of length zero, or of zero homology and of length 1.