This is from the book: Hilton and Stammbach, A Course in Homological Algebra, Chapter IV, Derived Functors, exercise 4.2.
Let $\varphi:C \to D$ be a chain map of the projective complex $C$ into the projective complex $D$ with $C_n=D_n=0$, $n<0$. If $H(\varphi):H(C)\to H(D)$ is an isomorphism, then $\varphi:C \to D$ is an homotopy equivalence. (This is what I want to prove.)
I have to use that if $P$ is projective with $P_n=0$, $n<0$, then $H(P)=0$ if and only if $1\simeq0:P \to P$, and also use the mapping cone $E(\varphi)$ of the chain map where $E_n=C_{n-1} \oplus D_n$ and $∂(a,b)=(-∂a,δa+∂b)$, $a \in C_{n-1}$, $b\in D_n$ to prove it.
Theorem 4.1.
Let $C: \dots \to C_n \to C_{n-1} \to \dots \to C_0$ be a projective complex and let $D: \dots \to D_n \to D_{n-1} \to \dots \to D_0$ be acyclic. Then there exists, for every homomorphism $\varphi:H_0(C) \to H_0(D)$, a chain map $\varphi':C \to D$ inducing $\varphi$.
Moreover, two chain maps inducing $\varphi$ are homotopic.
With this theorem I proved that if $P$ is projective with $P_n=0$, $n<0$, then $H(P)=0$ if and only if $1\simeq0:P \to P$.
Can you help me prove this? Thank you.
As you probably know the homology of the mapping cone is related to the homotopy of the chain complexes $C$ and $D$ by the following exact sequence:
$$ \cdots\to H_n(D)\to H_n(E(\varphi))\to H_{n-1}(C)\stackrel{\varphi_*}\to H_{n-1}(D)\to\cdots $$
If $\varphi_*:H(C)\to H(D)$ is an isomorphism, then $H(E(\varphi))=0$. Since $E(\varphi)$ is projective (in your case) we get (from exercise 4.1) that $1\simeq 0:E(\varphi)\to E(\varphi)$.
Now we have to prove that $\varphi$ is a homotopy equivalence. Since $E_n=C_{n-1}\oplus D_n$ we can write the differential as a matrix $\left(\begin{array}{cc} -\partial_C & 0\\ \varphi & \partial_D \end{array}\right).$ Let $s$ be a chain homotopy between $1$ and $0$. We also can write $s$ as a matrix $\left(\begin{array}{cc} s_1 & \psi\\ s_2 & s_3 \end{array}\right).$ Then $\psi$ is a chain map, $s_1$ is a chain homotopy from $\psi\varphi$ to $1_C$, and $s_3$ is a chain homotopy from $1_D$ to $\varphi\psi$, so $\varphi$ is a homotopy equivalence.