Let $F:A \to B$ be an additive left-exact functor of abelian categories (Do not assume that they have enough injectives / projectives.) Suppose we are given a class of objects $R$ adapted to $F$ (see below for definition). Is there a closed model category structure on $Com^+(A)$ such that weak equivalences are quasi-isomorphisms and $R$ is (or contains) the class of cofibrant objects?
Here is a definition of class of objects adapted to a left exact functor $F: A \to B$ (from Gelfand and Manin): it is a class of objects $R$ stable under direct sums, $F$ maps acyclic complexes from $Com^+(R)$ to acyclic ones and each object is in $A$ is a subobject of an object in $R$
(1) This question originated from this one: http://mathoverflow.net/search?q=adapted+class+of+objects+model+categories.
(2) This question does not really have anything to do with algebraic geometry, but I thought that algebraic geometers may have something to say about this.