Hello!


I'm a student learning the basics of working with the unbounded derived category $D(\mathcal{A})$. I arrived at the natural question, "is every K-projective complex formed out of projective objects?" (having learned that this isn't sufficient to guarantee K-projectivity), and a literature hunt led to the introduction of the paper "The Direct Limit Closure of Perfect Complexes" by Christensen and Holm (<http://arxiv.org/pdf/1301.0731.pdf>), where the authors mention K-projective and formed out of projective objects is equivalent to DG-projective (in the first paragraph).


I think I managed to prove this, but I still have two questions:


i. I gather that this last statement can be found in Avramov-Foxby-Halperin's Differential Graded Homological Algebra, but I couldn't locate a copy of this on the web. Does anyone know where I could obtain a copy of this (preferably digital)?


ii. With the result, it seems like there should be complexes which are K-projective but not DG-projective, but I have been unable to build one. Is there a reference where one is built/does anyone know of such an example (my hunch is that Avramov-Foxby-Halperin should be said reference)?


Many thanks! 



DG-projective vs. K-projective complexes
K-projectivity of a complex is a property of its homotopy equivalence class, i.e., any complex homotopy equivalent to a K-projective complex is K-projective. In particular, any contractible complex is K-projective. Taking the cone of the identity endomorphism of any complex that is not formed out of projective objects thus provides an example of a K-projective complex that is not formed out of projective objects.


