Hom-computable coalgebras, a composition factors matrix and the Euler bilinear form of an Euler coalgebra

A composition factors matrix is studied for any basic Hom-computable K-coalgebra C over an arbitrary field K, in connection with a Cartan matrix of C. Left Euler K-coalgebras C are defined. They are studied by means of an Euler integral bilinear form , the Euler characteristic χC(M,N) of Euler pairs of C-comodules M and N, and an Euler defect of C. It is shown that bC(lgthM,lgthN)=χC(M,N)+∂C(M,N), for all M, N in C-comod, and ∂C=0, if all simple C-comodules are of finite injective dimension. A diagrammatic characterisation of representation-directed hereditary Hom-computable coalgebras is given.