  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.

Representation Theory;K-Theory and Homology