Non-orbicular modules for Galois coverings

Piotr Dowbor. "Non-orbicular modules for Galois coverings." Colloquium Mathematicae 89.2 (2001): 241-310. <http://eudml.org/doc/284031>.

@article{PiotrDowbor2001,
abstract = {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 $Φ^\{B\} : Iₙ - spr(H) → mod(R/G)$, 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 B̃ with the same stabilizer, then usually the subcategory of non-orbicular indecomposables in $mod_\{B̃,B\}(R/G)$ 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 B̃ ≃ B for any infinite G-atom B with $End_\{R\}(B)/J(End_\{R\}(B)) ≃ k$ (Theorem 7.1). For this purpose the techniques of neighbourhoods (Theorem 7.2) and extension embeddings for matrix rings (Theorem 6.3) are developed.},
author = {Piotr Dowbor},
journal = {Colloquium Mathematicae},
keywords = {Galois coverings; tame representation type; orbicular modules; groups of automorphisms; locally bounded categories; covering functors},
language = {eng},
number = {2},
pages = {241-310},
title = {Non-orbicular modules for Galois coverings},
url = {http://eudml.org/doc/284031},
volume = {89},
year = {2001},
}

TY - JOUR
AU - Piotr Dowbor
TI - Non-orbicular modules for Galois coverings
JO - Colloquium Mathematicae
PY - 2001
VL - 89
IS - 2
SP - 241
EP - 310
AB - 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 $Φ^{B} : Iₙ - spr(H) → mod(R/G)$, 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 B̃ with the same stabilizer, then usually the subcategory of non-orbicular indecomposables in $mod_{B̃,B}(R/G)$ 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 B̃ ≃ B for any infinite G-atom B with $End_{R}(B)/J(End_{R}(B)) ≃ k$ (Theorem 7.1). For this purpose the techniques of neighbourhoods (Theorem 7.2) and extension embeddings for matrix rings (Theorem 6.3) are developed.
LA - eng
KW - Galois coverings; tame representation type; orbicular modules; groups of automorphisms; locally bounded categories; covering functors
UR - http://eudml.org/doc/284031
ER -