Tame comodule type, roiter bocses, and a geometry context for coalgebras

  • D. Simson

Abstract

We study the class of coalgebras $C$ of $fc$-tame comodule type introduced by the author. With any basic computable $K$-coalgebra $C$ and a bipartite vector $v = (v′|v″) ∈ K_0(C) × K_0(C)$, we associate a bimodule matrix problem $\textbf{Mat}^v_C(ℍ)$, an additive Roiter bocs $\textbf{B}^C_v$, an affine algebraic $K$-variety $\textbf{Comod}^C_v$, and an algebraic group action $\textbf{G}^C_v × \textbf{Comod}^C_v → \textbf{Comod}^C_v$. We study the $fc$-tame comodule type and the fc-wild comodule type of $C$ by means of $\textbf{Mat}^v_C(ℍ)$, the category $\textbf{rep}_K (\textbf{B}^C_v)$ of $K$-linear representations of $\textbf{B}^C_v$, and geometry of $\textbf{G}^C_v$ -orbits of $\textbf{Comod}_v$. For computable coalgebras $C$ over an algebraically closed field $K$, we give an alternative proof of the $fc$-tame-wild dichotomy theorem. A characterization of $fc$-tameness of $C$ is given in terms of geometry of $\textbf{G}^C_v$-orbits of $\textbf{Comod}^C_v$. In particular, we show that $C$ is $fc$-tame of discrete comodule type if and only if the number of $\textbf{G}^C_v$-orbits in $\textbf{Comod}^C_v$ is finite for every $v = (v′|v″) ∈ K_0(C) × K_0(C)$.
Published
25.06.2009
How to Cite
SimsonD. “Tame Comodule Type, Roiter Bocses, and a Geometry Context for Coalgebras”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 61, no. 6, June 2009, pp. 810-33, http://umj.imath.kiev.ua/index.php/umj/article/view/3060.
Section
Research articles