If $h\_i:A\_i\to A\_{i+1}$ is a countable chain of morphisms in an abelian category $A$ that is AB3 then one can consider the (Bökstedt-Neeman) homotopy colimit of $A\_i$ in $D^b(A)$. This is a two-term complex $\coprod A\_i\stackrel{f}{\to} \coprod A\_i$ with the corresponding "components" of $f$ being $id\_{A\_i}$, $-h\_i$, and $0$, respectively. Clearly, the cokernel of $f$ is the colimit of $A\_i$. My question is: which conditions can ensure that $f$ is a monomorphism? 


$f$ is a colimit of monomorphisms; so it is a monomorphism if $A$ is AB5. Is $f$ monomorphic for $A$ being a general AB3 category (what about AB4?:))? Are there any alternative methods to study the kernel of $f$? Are there any references for this problem?


Upd. So, a nice example of prof. Rickard demonstrates that AB4 is not sufficient. However, I am (still) interested in "non-standard axioms" for $A$ that would ensure the injectivity of $f$.



On countable homotopy colimits in (the derived categories of) AB3 abelian categories
$\textrm{AB4}$ (and *a fortiori* $\textrm{AB3}$) is not enough, as it's not true for the opposite category of a module category, which is $\textrm{AB4}$.


Let $R$ be any ring, and consider the inverse system
$$\dots\to R[x]\stackrel{\theta}{\to}R[x]\stackrel{\theta}{\to}R[x]\stackrel{\theta}{\to}R[x],$$
of $R$-modules, where the maps $\theta$ are multiplication by $x$.


The map
$$\prod R[X]\stackrel{\text{id}-{\theta}}{\longrightarrow}\prod R[x]$$
is not surjective, as $(1,1,1,\dots)$ is not in the image.


