INCIDENCE COALGEBRAS OF INTERVALLY FINITE POSETS, THEIR INTEGRAL QUADRATIC FORMS AND COMODULE CATEGORIES

The incidence coalgebras C = (KI)-I-square of intervally finite posets I and their comodules are studied by means of their Cartan matrices and the Euler integral bilinear form b(C) : Z((I)) x Z((I)) -> Z. One of our main results asserts that, under a suitable assumption on I, C is an Euler coalgebra with the Euler defect partial derivative(C) : D-(I) x Z((I)) -> Z zero and b(C) (lgth M, lgth N) = chi(C) (M, N) for any pair of indecomposable left C-comodules M and N of finite K-dimension, where xc (M, N) is the Euler characteristic of the pair M, N and lgth M is an element of Z((I)) is the composition length vector. The structure of minimal injective resolutions of simple left C-comodules is described by means of the inverse C-I(1) is an element of M-I(<=)(Z) of the incidence matrix C-I is an element of M-I(Z) of the poset I. Moreover, we describe the Bass numbers mu(I)(m)(S-I(a), S-I(b)), with m >= 0, for any simple (KI)-I-square-comodules S-I(a), S-I(b) by means of the coefficients of the bth row of C-I(-1). We also show that, for any poset I of width two, the Grothendieck group K-0((KI)-I-square-Comod(fc)) of the category of finitely copresented (KI)-I-square-comodules is generated by the classes [Si (a)] of the simple comodules S-I(a) and the classes [E-I(a)] of the injective covers E-I(a) of S-I(a), with a is an element of I.