Incidence coalgebras of interval finite posets of tame comodule type

• Autorzy: Zbigniew Leszczyński , Daniel Simson
• Języki publikacji: EN

Abstrakty

EN

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 𝔸̃ *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 𝔸̃ *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.

Kategorie tematyczne

• 16G20: Representations of quivers and partially ordered sets
• 16W80: Topological and ordered rings and modules
• 16G60: Representation type (finite, tame, wild, etc.)

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2015
• Tom: 141
• Numer: 2
• Strony: 261-295

Daty

wydano

2015

Twórcy

autor

Zbigniew Leszczyński

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

autor

Daniel Simson

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

Identyfikatory

DOI

10.4064/cm141-2-10