I am not sure where one looks up this type of fact. Google was not very helpful.



Do unbounded chain complexes have enough injectives?
Yes, any Grothendieck abelian category has enough injectives. I believe this goes back to Grothendieck's Tohoku paper. The category in question is Grothendieck abelian since it is equivalent to a category of additive presheaves. The domain category has objects the integers, and morphisms generated by $d\_n: n \rightarrow n+1$ (or the other way around depending on your grading conventions), subject to the relations $d\_{n+1} \circ d\_n=0$.


