TY - JOUR
AU - D. Simson
PY - 2009/06/25
Y2 - 2024/08/06
TI - Tame comodule type, roiter bocses, and a geometry context for coalgebras
JF - Ukrains’kyi Matematychnyi Zhurnal
JA - Ukr. Mat. Zhurn.
VL - 61
IS - 6
SE - Research articles
DO -
UR - https://umj.imath.kiev.ua/index.php/umj/article/view/3060
AB - 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 $v = (v′|v″) ∈ K_0(C) × K_0(C)$.
ER -