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

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

Zbl 1063.16017
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