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

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 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 C 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 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).