Flat covers of representations of the quiver

Rooted quivers are quivers that do not contain A∞≡⋯→•→• as a subquiver. The existence of flat covers and cotorsion envelopes for representations of these quivers have been studied by Enochs et al. The main goal of this paper is to prove that flat covers and cotorsion envelopes exist for representations of A∞. We first characterize finitely generated projective representations of A∞. We also see that there are no projective covers for representations of A∞, which adds more interest to the problem of the existence of flat covers.