Non-orbicular modules for Galois coverings

Piotr Dowbor¹

¹ Faculty of Mathematics and Computer Science Nicholas Copernicus University Chopina 12/18 87-100 Toru/n, Poland

Colloquium Mathematicum, Tome 89 (2001) no. 2, pp. 241-310

Résumé

Given a group $G$ of $k$-linear automorphisms of a locally bounded $k$-category $R$, the problem of existence and construction of non-orbicular indecomposable $R/G$-modules is studied. For a suitable finite sequence $B$ of $G$-atoms with a common stabilizer $H$, a representation embedding ${\mit \Phi }^{B} : \mathop {I_n\hbox {\rm-spr}}\nolimits (H)\to \mathop {\hbox {mod}}(R/G)$, which yields large families of non-orbicular indecomposable $R/G$-modules, is constructed (Theorem 3.1). It is proved that if a $G$-atom $B$ with infinite cyclic stabilizer admits a non-trivial left Kan extension $\widetilde {\! B}$ with the same stabilizer, then usually the subcategory of non-orbicular indecomposables in $\mathop {\hbox {mod}}_{\{ \widetilde {B},B\} }(R/G)$ is wild (Theorem 4.1, also 4.5). The analogous problem for the case of different stabilizers is discussed in Theorem 5.5. It is also shown that if $R$ is tame then $\widetilde {B}\simeq B$ for any infinite $G$-atom $B$ with $\mathop {\hbox {End}}_R(B)/J(\mathop {\hbox {End}}_R(B)) \simeq k$ (Theorem 7.1). For this purpose the techniques of neighbourhoods (Theorem 7.2) and extension embeddings for matrix rings (Theorem 6.3) are developed.

Détail

DOI : 10.4064/cm89-2-9

Keywords: given group k linear automorphisms locally bounded k category problem existence construction non orbicular indecomposable g modules studied suitable finite sequence g atoms common stabilizer representation embedding mit phi mathop hbox rm spr nolimits mathop hbox mod which yields large families non orbicular indecomposable g modules constructed theorem proved g atom infinite cyclic stabilizer admits non trivial kan extension widetilde stabilizer usually subcategory non orbicular indecomposables mathop hbox mod widetilde wild theorem analogous problem different stabilizers discussed theorem shown tame widetilde simeq infinite g atom mathop hbox end mathop hbox end simeq theorem purpose techniques neighbourhoods theorem extension embeddings matrix rings theorem developed