Tame comodule type, roiter bocses, and a geometry context for coalgebras - Simson, D. (Ukrains’kyi Matematychnyi Zhurnal, 2009)
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 $Mat^v_C(ℍ)$, an additive Roiter bocs $B^C_v$, an affine algebraic $K$-variety $Comod^C_v$, and an algebraic group action $G^C_v × Comod^C_v → Comod^C_v$. We study the $fc$-tame comodule type and the fc-wild comodule type of $C$ by means of $Mat^v_C(ℍ)$, the category $rep_K (B^C_v)$ of $K$-linear representations of $B^C_v$, and geometry of $G^C_v$ -orbits of $Comod_v$. 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 $G^C_v$-orbits of $Comod^C_v$. In particular, we show that $C$ is $fc$-tame of discrete comodule type if and only if the number of $G^C_v$-orbits in $Comod^C_v$ is finite for every $v = (v′|v″) ∈ K_0(C) × K_0(C)$.