On artin algebras with almost all indecomposable modules of projective or injective dimension at most one

Andrzej Skowroński

Open Mathematics (2003)

  • Volume: 1, Issue: 1, page 108-122
  • ISSN: 2391-5455

Abstract

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Let A be an artin algebra over a commutative artin ring R and ind A the category of indecomposable finitely generated right A-modules. Denote ℒ A to be the full subcategory of ind A formed by the modules X whose all predecessors in ind A have projective dimension at most one, and by ℛ A the full subcategory of ind A formed by the modules X whose all successors in ind A have injective dimension at most one. Recently, two classes of artin algebras A with ℒ A ∪ ℛ A co-finite in ind A, quasi-tilted algebras and generalized double tilted algebras, have been extensively investigated. The aim of the paper is to show that these two classes of algebras exhaust the class of all artin algebras A for which ℒ A ∪ ℛ A is co-finite in ind A, and derive some consequences.

How to cite

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Andrzej Skowroński. "On artin algebras with almost all indecomposable modules of projective or injective dimension at most one." Open Mathematics 1.1 (2003): 108-122. <http://eudml.org/doc/268914>.

@article{AndrzejSkowroński2003,
abstract = {Let A be an artin algebra over a commutative artin ring R and ind A the category of indecomposable finitely generated right A-modules. Denote \[\mathcal \{L\}\_A\] to be the full subcategory of ind A formed by the modules X whose all predecessors in ind A have projective dimension at most one, and by \[\mathcal \{R\}\_A\] the full subcategory of ind A formed by the modules X whose all successors in ind A have injective dimension at most one. Recently, two classes of artin algebras A with \[\mathcal \{L\}\_A \cup \mathcal \{R\}\_A\] co-finite in ind A, quasi-tilted algebras and generalized double tilted algebras, have been extensively investigated. The aim of the paper is to show that these two classes of algebras exhaust the class of all artin algebras A for which \[\mathcal \{L\}\_A \cup \mathcal \{R\}\_A\] is co-finite in ind A, and derive some consequences.},
author = {Andrzej Skowroński},
journal = {Open Mathematics},
keywords = {Primary 16G70; 18G20; Secondary 16G10},
language = {eng},
number = {1},
pages = {108-122},
title = {On artin algebras with almost all indecomposable modules of projective or injective dimension at most one},
url = {http://eudml.org/doc/268914},
volume = {1},
year = {2003},
}

TY - JOUR
AU - Andrzej Skowroński
TI - On artin algebras with almost all indecomposable modules of projective or injective dimension at most one
JO - Open Mathematics
PY - 2003
VL - 1
IS - 1
SP - 108
EP - 122
AB - Let A be an artin algebra over a commutative artin ring R and ind A the category of indecomposable finitely generated right A-modules. Denote \[\mathcal {L}_A\] to be the full subcategory of ind A formed by the modules X whose all predecessors in ind A have projective dimension at most one, and by \[\mathcal {R}_A\] the full subcategory of ind A formed by the modules X whose all successors in ind A have injective dimension at most one. Recently, two classes of artin algebras A with \[\mathcal {L}_A \cup \mathcal {R}_A\] co-finite in ind A, quasi-tilted algebras and generalized double tilted algebras, have been extensively investigated. The aim of the paper is to show that these two classes of algebras exhaust the class of all artin algebras A for which \[\mathcal {L}_A \cup \mathcal {R}_A\] is co-finite in ind A, and derive some consequences.
LA - eng
KW - Primary 16G70; 18G20; Secondary 16G10
UR - http://eudml.org/doc/268914
ER -

References

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