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A Homotopy of Quiver Morphisms with Applications to Representations

Published online by Cambridge University Press:  20 November 2018

Edgar E. Enochs  and
Ivo Herzog
Edgar E. Enochs
Affiliation:
Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506-0027, U.S.A.
Ivo Herzog
Affiliation:
Department of Mathematics, The Ohio State University at Lima, Lima, Ohio 45804, U.S.A.
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Abstract

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It is shown that a morphism of quivers having a certain path lifting property has a decomposition that mimics the decomposition of maps of topological spaces into homotopy equivalences composed with fibrations. Such a decomposition enables one to describe the right adjoint of the restriction of the representation functor along a morphism of quivers having this path lifting property. These right adjoint functors are used to construct injective representations of quivers. As an application, the injective representations of the cyclic quivers are classified when the base ring is left noetherian. In particular, the indecomposable injective representations are described in terms of the injective indecomposable $R$-modules and the injective indecomposable $R[\text{x},\,{{\text{x}}^{-1}}]$ -modules.

Keywords

18A4016599
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 51 , Issue 2 , 01 April 1999 , pp. 294 - 308
Copyright
Copyright © Canadian Mathematical Society 1999

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