Homotopy Category of Cotorsion Flat Representations of Quivers

Recently in [10], it was proved that over any ring R, there exists a complete cotorsion pair (Kp(Flat-R); K(dg-CotF-R)) in K(Flat-R), the homotopy category of complexes of flat R-modules, where Kp(Flat-R) and K(dg-CotF-R) are the homotopy categories raised by flat (or pure) and dgcotorsion complexes of flat R-modules, respectively. This paper aims at recognition of a parallel cotorsion pair in K(Flat-Q), the homotopy category of flat representation of certain quivers Q, where Q may also be infinite. The importance of this result lies in the fact that this homotopy categories do not necessarily raise from the category of modules over some ring. In the other part of this paper, we give a classification of compact objects in K(dg-CotF-Q), the homotopy category of dg-cotorsion complexes of flat representations of certain Q, in terms of the corresponding vertex-complexes