Flat covers of representations of the quiver \(A_\infty\). (English) Zbl 1066.16011

Enochs, E.; Estrada, S.; García Rozas, J. R.; Oyonarte, L.

Int. J. Math. Math. Sci. 2003, No. 70, 4409-4419 (2003).

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 E. Enochs, L. Oyonarte and 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.

MSC:

16G20	Representations of quivers and partially ordered sets
16D40	Free, projective, and flat modules and ideals in associative algebras

Keywords:

rooted quivers; flat covers; cotorsion envelopes; representations of quivers; finitely generated projective representations

Citations:

Zbl 1063.16017