Coxeter transformation and inverses of Cartan matrices for coalgebras

Let C be a coalgebra and consider the Grothendieck groups of the categories of the socle-finite injective right and left C-comodules. One of the main aims of the paper is to study Coxeter transformation, and its dual, of a pointed sharp Euler coalgebra C, and to relate the action of these transformations on a class of indecomposable finitely cogenerated C-comodules N with almost split sequences starting or ending with N. We also show that if C is a pointed K-coalgebra such that the every vertex of the left Gabriel quiver 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 of finitely cogenerated left C-comodules ending at N. We show that the dimension vector of the Auslander-Reiten translate given by the Coxeter transformation, if C is hereditary, or more generally, if inj.dim DN=1 and Hom(C,DN)=0.