2017
Том 69
№ 9

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

Simson D.

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 $v = (v′|v″) ∈ K_0(C) × K_0(C)$.

English version (Springer): Ukrainian Mathematical Journal 61 (2009), no. 6, pp 964-987.

Citation Example: Simson D. Tame comodule type, roiter bocses, and a geometry context for coalgebras // Ukr. Mat. Zh. - 2009. - 61, № 6. - pp. 810-833.

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