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

Authors

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

$v = (v′|v″) ∈ K_0(C) × K_0(C)$ .