The Euler characteristic and Euler defect for comodules over Euler coalgebras

Let K be a field. We study a class of left C-comodules over a basic left Euler coalgebra C by means of the Euler Z-bilinear form (b) over cap (C) associated to C, the Euler characteristic chi(C)(M, N) of left C-comodules M, N, and the defect partial derivative(C) (M, N) is an element of Z associated to any computable Euler pair (M, N) of left C-comodules. We show that (b) over cap (C)(lgthM, lgthN) = chi(C)(M, N) + partial derivative(C)(M, N), for any computable Euler pair (M, N) of comodules over a left Euler coalgebra C. One of the main results of the paper asserts that the defect partial derivative(C)(M, N) is zero and (b) over cap (C)(lgthM, lgthN) = chi(C)(M, N), if the comodules M, N are finite-dimensional.