Incidence coalgebras of interval finite posets of tame comodule type

Zbigniew Leszczyński ; Daniel Simson

Colloquium Mathematicae, Tome 139 (2015), p. 261-295 / Harvested from The Polish Digital Mathematics Library

Résumé

The incidence coalgebras $K^{□} I$ of interval finite posets I and their comodules are studied by means of the reduced Euler integral quadratic form $q^{•}: ℤ^{(I)} → ℤ$, where K is an algebraically closed field. It is shown that for any such coalgebra the tameness of the category $K^{□} I-comod$ of finite-dimensional left $K^{□} I$-modules is equivalent to the tameness of the category $K^{□} I-Comod_{fc}$ of finitely copresented left $K^{□} I$-modules. Hence, the tame-wild dichotomy for the coalgebras $K^{□} I$ is deduced. Moreover, we prove that for an interval finite ̃ *ₘ-free poset I the incidence coalgebra $K^{□} I$ is of tame comodule type if and only if the quadratic form $q^{•}$ is weakly non-negative. Finally, we give a complete list of all infinite connected interval finite ̃ *ₘ-free posets I such that $K^{□} I$ is of tame comodule type. In this case we prove that, for any pair of finite-dimensional left $K^{□} I$-comodules M and N, $b̅_{K^{□} I} (dim M,dim N) = ∑_{j=0}^{∞} (-1)^{j} dim_{K} Ext_{K^{□} I}^{j}(M,N)$, where $b̅_{K^{□} I}: ℤ^{(I)} × ℤ^{(I)} → ℤ$ is the Euler ℤ-bilinear form of I and dim M, dim N are the dimension vectors of M and N.

Détail

Publié le : 2015-01-01

Zbl 1339.16018

EUDML-ID : urn:eudml:doc:284146