Flat covers of representations of the quiver \(A_\infty\). (Q1774367)

Summary: Rooted quivers are quivers that do not contain \(A_\infty\equiv\cdots\to\bullet\to\bullet\) as a subquiver. The existence of flat covers and cotorsion envelopes for representations of these quivers have been studied by \textit{E. Enochs, L. Oyonarte} and \textit{B. Torrecillas} [Commun. Algebra 32, No. 4, 1319-1338 (2004; Zbl 1063.16017)]. The main goal of this paper is to prove that flat covers and cotorsion envelopes exist for representations of \(A_\infty\). We first characterize finitely generated projective representations of \(A_\infty\). We also see that there are no projective covers for representations of \(A_\infty\), which adds more interest to the problem of the existence of flat covers.