Coxeter transformation and inverses of Cartan matrices for coalgebras

• Published in: Journal of algebra 2010-11, Vol.324 (9), p.2223-2248
• Main Authors: Chin, William, Simson, Daniel
• Format: Article
• Language: English
• Subjects: Auslander–Reiten quiver; Cartan matrix; Coalgebra; Comodule; Coxeter transformation; Pseudocompact algebra
• ISSN: 0021-8693; 1090-266X
• DOI: 10.1016/j.jalgebra.2010.06.029

Let C be a coalgebra and let Z ► I C , Z ◂ I C ⊆ Z I C be the Grothendieck groups of the category C o p -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 Φ C : Z ► I C → Z ◂ I C and its dual Φ C − : Z ◂ I C → Z ► I C of a pointed sharp Euler coalgebra C, and to relate the action of Φ C and Φ C − 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 Q C 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 → τ C N → N ′ → N → 0 in the category C - Comod f c of finitely cogenerated left C-comodules, with an indecomposable comodule τ C N . We show that dim τ C N = Φ C ( dim N ) , if C is hereditary, or more generally, if inj.dim D N = 1 and Hom C ( C , D N ) = 0 .