Incidence coalgebras of intervally finite posets, their integral quadratic forms and comodule categories

Daniel Simson¹

¹ Faculty of Mathematics and Computer Science Nicolaus Copernicus University Chopina 12/18 87-100 Toruń, Poland

Colloquium Mathematicum, Tome 115 (2009) no. 2, pp. 259-295

Résumé

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 $a\in I$.

Détail

DOI : 10.4064/cm115-2-9

Keywords: incidence coalgebras square intervally finite posets their comodules studied means their cartan matrices euler integral bilinear form mathbb times mathbb rightarrow mathbb main results asserts under suitable assumption nbsp nbsp euler coalgebra euler defect partial mathbb times mathbb rightarrow mathbb zero lgth lgth chi pair indecomposable c comodules finite k dimension where chi euler characteristic pair nbsp lgth mathbb composition length vector structure minimal injective resolutions simple c comodules described means inverse mathfrak mathbb preceq mathbb incidence matrix mathfrak mathbb mathbb poset nbsp moreover describe bass numbers i geq simple square i comodules means coefficients bth row mathfrak poset width grothendieck group square hbox comod category finitely copresented square i comodules generated classes simple comodules classes injective covers