Injective model structure on sheaves of bounded complexes of $A$-modules

The following might be very well known for people who works with model categories, but I do not find the answer.

Let $A$-be a ring. Denote $\mathbf{Ch}_+(A)$ the category of positive degree cochain complexes of $A$-modules (complexes where differential rises the degree and which are zero in negative degree). There is the injective model structure on $\mathbf{Ch}_+(A)$ which has quasi-isomorphisms as weak equivalences, monomorphisms as cofibrations and epimorphisms with injective kernel as fibrations. Let $K^\bullet$ be a complex, then its injective resolution $I^\bullet (K^\bullet)$ turn out to be its fibrant replacement.

Let $X$ be a topological space and consider $\mathbf{Ch}_+(\mathbf{Shv}_X(A))$, the category of positive complexes of sheaves of $A$-modules on $X$. My question is the following:

Question: Is there an analogue model structure in $\mathbf{Ch}_+(\mathbf{Shv}_X(A))$ to the injective model structure in $\mathbf{Ch}_+(A)$? More concretely, if we consider quasi-isomorphisms as weak equivalences, monomorphisms as cofibrations, and morphisms having the right lifting property with respect to injective quasi-isomorphisms, do they define a model structure? If $\mathcal{K^\bullet}$ is a complex of sheaves, will its injective resolution $I^\bullet (\mathcal{K}^\bullet)$ be its fibrant replacement?

Thank you very much

• I don't have time to write a full answer just now, but I think the answer is probably yes, and that proof should use cotorsion pairs. I recommend you look at the work of Hovey and Gillespie. Gillespie has a paper handling categories of chain complexes of quasi-coherent sheaves over a scheme, and perhaps some mention is made there of the case you care about. I believe there are two cotorsion pairs in these settings, one for the projective model structure and one for the injective. – David White Aug 30 '15 at 22:22
• I think that, in all cases, cofibrations should be cochain maps which are mono in strictly positive degrees (not necessarily in degree $0$), so that the $t$-structure truncation functor from the unbounded category is a left Quillen functor. – Fernando Muro Aug 31 '15 at 0:03

Question: Is there an analogue model structure in $\mathbf{Ch}_+(\mathbf{Shv}_X(A))$ to the injective model structure on $\mathbf{Ch}_+(A)$? More concretely, if we consider quasi-isomorphisms as weak equivalences, monomorphisms as cofibrations, and morphisms having the right lifting property with respect to injective quasi-isomorphisms, do they define a model structure?
Question: If $\mathcal{K}^\bullet$ is a complex of sheaves, will its injective resolution $I^\bullet(\mathcal{K}^\bullet)$ be its fibrant replacement?