Properties of G-atoms and full Galois covering reduction to stabilizers

Piotr Dowbor

Colloquium Mathematicae (2000)

  • Volume: 83, Issue: 2, page 231-265
  • ISSN: 0010-1354

Abstract

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Given a group G of k-linear automorphisms of a locally bounded k-category R it is proved that the endomorphism algebra E n d R ( B ) of a G-atom B is a local semiprimary ring (Theorem 2.9); consequently, the injective E n d R ( B ) -module ( E n d R ( B ) ) * is indecomposable (Corollary 3.1) and the socle of the tensor product functor - ⊗ R B * is simple (Theorem 4.4). The problem when the Galois covering reduction to stabilizers with respect to a set U of periodic G-atoms (defined by the functors Φ U : ∐ B ∈ U m o d k G B → m o d ( R / G ) and Ψ U : m o d ( R / G ) → ∏ B ∈ U m o d k G B )is full (resp. strictly full) is studied (see Theorems A, B and 6.3).

How to cite

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Dowbor, Piotr. "Properties of G-atoms and full Galois covering reduction to stabilizers." Colloquium Mathematicae 83.2 (2000): 231-265. <http://eudml.org/doc/210784>.

@article{Dowbor2000,
abstract = {Given a group G of k-linear automorphisms of a locally bounded k-category R it is proved that the endomorphism algebra $End_R (B)$ of a G-atom B is a local semiprimary ring (Theorem 2.9); consequently, the injective $End_R (B)$-module $(End_R (B))^*$ is indecomposable (Corollary 3.1) and the socle of the tensor product functor $- ⊗_R B^*$ is simple (Theorem 4.4). The problem when the Galois covering reduction to stabilizers with respect to a set U of periodic G-atoms (defined by the functors $Φ^U: \coprod _\{B ∈ U\} mod kG_B → mod(R/G)$ and $Ψ^U: mod(R/G) → \prod _\{B ∈ U\} mod kG_B$)is full (resp. strictly full) is studied (see Theorems A, B and 6.3).},
author = {Dowbor, Piotr},
journal = {Colloquium Mathematicae},
keywords = {Galois covering; tame; locally finite-dimensional module; Galois coverings; atoms; stabilizers; locally finite dimensional modules; categories of modules; infinite representation type},
language = {eng},
number = {2},
pages = {231-265},
title = {Properties of G-atoms and full Galois covering reduction to stabilizers},
url = {http://eudml.org/doc/210784},
volume = {83},
year = {2000},
}

TY - JOUR
AU - Dowbor, Piotr
TI - Properties of G-atoms and full Galois covering reduction to stabilizers
JO - Colloquium Mathematicae
PY - 2000
VL - 83
IS - 2
SP - 231
EP - 265
AB - Given a group G of k-linear automorphisms of a locally bounded k-category R it is proved that the endomorphism algebra $End_R (B)$ of a G-atom B is a local semiprimary ring (Theorem 2.9); consequently, the injective $End_R (B)$-module $(End_R (B))^*$ is indecomposable (Corollary 3.1) and the socle of the tensor product functor $- ⊗_R B^*$ is simple (Theorem 4.4). The problem when the Galois covering reduction to stabilizers with respect to a set U of periodic G-atoms (defined by the functors $Φ^U: \coprod _{B ∈ U} mod kG_B → mod(R/G)$ and $Ψ^U: mod(R/G) → \prod _{B ∈ U} mod kG_B$)is full (resp. strictly full) is studied (see Theorems A, B and 6.3).
LA - eng
KW - Galois covering; tame; locally finite-dimensional module; Galois coverings; atoms; stabilizers; locally finite dimensional modules; categories of modules; infinite representation type
UR - http://eudml.org/doc/210784
ER -

References

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