Incidence coalgebras of intervally finite posets, their integral quadratic forms and comodule categories
Tom 115 / 2009
Streszczenie
The incidence coalgebras $C = K^{\square} I$ of intervally finite posets $I$ and their comodules are studied by means of their Cartan matrices and the Euler integral bilinear form $b_C:\mathbb Z^{(I)}\times\mathbb Z^{(I)}\rightarrow \mathbb Z$. One of our main results asserts that, under a suitable assumption on $I$, $C$ is an Euler coalgebra with the Euler defect $\partial_C:\mathbb Z^{(I)}\times\mathbb Z^{(I)}\rightarrow \mathbb Z$ zero and $b_C ({\bf lgth}\, M,{\bf lgth}\, N) =\chi_C(M,N) $ for any pair of indecomposable left $C$-comodules $M$ and $N$ of finite $K$-dimension, where $\chi_C(M,N)$ is the Euler characteristic of the pair $M$, $N$ and ${\bf lgth}\, M\in \mathbb Z^{(I)}$ is the composition length vector. The structure of minimal injective resolutions of simple left $C$-comodules is described by means of the inverse ${\mathfrak C}_I^{-1}\in {\mathbb M}^\preceq_I(\mathbb Z)$ of the incidence matrix ${\mathfrak C}_I \in {\mathbb M}_I(\mathbb Z)$ of the poset $I$. Moreover, we describe the Bass numbers $\mu^I_m(S_I(a), S_I(b))$, with $m\geq 0$, for any simple $K^{\square} I$-comodules $S_I(a)$, $S_I(b)$ by means of the coefficients of the $b$th row of ${\mathfrak C}_I^{-1}$. We also show that, for any poset $I$ of width two, the Grothendieck group ${\bf K}_0(K^{\square} I\hbox{-\rm Comod}_{\rm fc})$ of the category of finitely copresented $K^{\square} I$-comodules is generated by the classes $[S_I(a)]$ 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 $a\in I$.
