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Let C be a coalgebra and consider the Grothendieck groups of the categoriesof the socle-finite injective right and left C-comodules. One of the main aimsof the paper is to study Coxeter transformation, and its dual, of a pointedsharp Euler coalgebra C, and to relate the action of these transformations on aclass of indecomposable finitely cogenerated C-comodules N with almost splitsequences starting or ending with N. We also show that if C is a pointedK-coalgebra such that the every vertex of the left Gabriel quiver of C has onlyfinitely many neighbours, then for any indecomposable non-projective leftC-comodule N of finite K-dimension, there exists a unique almost split sequenceof finitely cogenerated left C-comodules ending at N. We show that thedimension vector of the Auslander-Reiten translate given by the Coxetertransformation, if C is hereditary, or more generally, if inj.dim DN=1 andHom(C,DN)=0.

Coxeter transformation and inverses of Cartan matrices for coalgebras