International Journal of Mathematics and Mathematical Sciences
Volume 2006 (2006), Article ID 47146, 16 pages
doi:10.1155/IJMMS/2006/47146
Irreducible morphisms, the Gabriel-valued quiver and
colocalizations for coalgebras
Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, ul. Chopina 12/18, Toruń 87-100, Poland
Received 22 December 2005; Revised 7 May 2006; Accepted 9 May 2006
Copyright © 2006 Daniel Simson. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Given a basic K-coalgebra C, we study the left Gabriel-valued quiver (QC,dC) of C by means of irreducible morphisms between indecomposable injective left C-comodules and by means of the powers radm of the radical rad of the category
C-inj of the socle-finite injective left C-comodules.
Connections between the valued quiver (QC,dC) of C and the valued quiver (QC¯,dC¯) of a colocalization coalgebra quotient fE:C→C¯ of C are established.