This is a sort of follow up to [this MO question](https://mathoverflow.net/questions/165825/postnikov-towers-in-bounded-t-structures).


Let $R$ be a ring (eventually with good properties) and $\mathrm{Chains}(R)$ be the category of chain complexes of $R$-modules (eventually bounded). One has the natural endofunctors $\tau\_{\leq n}$ and $\tau\_{\geq n}$ associated with the standard t-structure on $\mathrm{Chains}(R)$. 


What I'm wondering is the following. Is it true that for every morphism $f\colon A\to B$ of chain complexes there exist morphisms $h:A\to C$ and $g:C\to B$ with $\tau\_{\leq -1}(h)\colon \tau\_{\leq -1}(A)\to \tau\_{\leq -1}(C)$ a weak equivalence (in the standard model structure on chain complexes), $\tau\_{\geq 0}(g)\colon \tau\_{\geq 0}(C)\to \tau\_{\geq 0}(B)$ a weak equivalence, and with $f$ homotopy equivalent to $g\circ h$?



Homotopy factorization of morphisms of chain complexes
Consider the following diagram in which the rows are cofiber sequences:
$$\require{AMScd}\begin{CD}
\tau\_{\geq0}A @>>> A @>>> \tau\_{\leq-1}A @>>>\Sigma\tau\_{\geq0}A\\
@VVV @VVV @| @VVV\\
\tau\_{\geq0}B @>>> C @>>> \tau\_{\leq-1}A @>>>\Sigma\tau\_{\geq0}B\\
@| @VVV @VVV @|\\
\tau\_{\geq0}B @>>> B @>>> \tau\_{\leq-1}B @>>>\Sigma\tau\_{\geq0}B
\end{CD}$$


Here the vertical maps in the first, third, and fourth columns are the maps induced by $f$, and $C$ can be obtained as the fiber of the rightmost map in the middle row. The second column is then the factorization you seek.


