On the representation type of locally bounded categories
Let K be an algebraically closed field. Given a K-algebra R we denote by mod(R) the category of finite dimensional right R-modules. We recall that R is representation-tame if for every \(d>0\) there are K[T]-R-bimodules \(_{K[T]}Q_ R^{(j)}\), \(j=1,...,m_ d\), such that: (i) \(_{K[T]}Q^{(j)}\) is finitely generated free, (ii) all but a finite number of indecomposable modules M in mod(R) with \(\dim_ kM=d\) are of the form \(M=S\otimes_{K[T]}Q_ R^{(j)}\) for some \(j\leq m_ d\) and a simple K[T]-module S. The paper contains interesting and useful properties equivalent to the tameness. The definition of representation-tame algebra is extended to locally bounded K-categories \(\tilde R\) and it is proved that \(\tilde R\) is representation-tame if and only if all finite full subcategories of \(\tilde R\) are representation-tame. The result is very useful in studying representation-tame algebras in terms of their covering categories (which are locally bounded).
category of finite dimensional right R-modules
scientific article; zbMATH DE number 3975152