Coxeter transformation and inverses of Cartan matrices for coalgebras

Abstract

Let C be a coalgebra and let Z►IC,Z◂IC⊆ZIC be the Grothendieck groups of the category Cop-inj and C-inj of the socle-finite injective right and left C-comodules, respectively. One of the main aims of the paper is to study the Coxeter transformation ΦC:Z►IC→Z◂IC and its dual ΦC−:Z◂IC→Z►IC of a pointed sharp Euler coalgebra C, and to relate the action of ΦC and ΦC− on a class of indecomposable finitely cogenerated C-comodules N with the ends of almost split sequences starting with N or ending at N. By applying Chin, Kleiner, and Quinn (2002) [5], we also show that if C is a pointed K-coalgebra such that the every vertex of the left Gabriel quiver QC of C has only finitely many neighbours then for any indecomposable non-projective left C-comodule N of finite K-dimension, there exists a unique almost split sequence 0→τCN→N′→N→0 in the category C-Comodfc of finitely cogenerated left C-comodules, with an indecomposable comodule τCN. We show that dimτCN=ΦC(dimN), if C is hereditary, or more generally, if inj.dimDN=1 and HomC(C,DN)=0.