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).