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

Open Mathematics (2003)

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

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topAndrzej 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

top- [1] M. Auslander, I. Reiten and S.O. Smalø, “Representation Theory of Artin Algebras”, Cambridge Studies in Advanced Mathematics, Vol. 36, Cambridge University Press, 1995. Zbl0834.16001
- [2] F. U. Coelho and M. A. Lanzilotta, Algebras with small homological dimension, Manuscripta Math. 100 (1999), 1–11. http://dx.doi.org/10.1007/s002290050191 Zbl0966.16001
- [3] F. U. Coelho and M. A. Lanzilotta, Weakly shod algebras, Preprint, Sao Paulo 2001. Zbl1062.16018
- [4] F. U. Coelho and A. Skowroński, On Auslander-Reiten components of quasi-tilted algebras, Fund. Math. 143 (1996), 67–82.
- [5] D. Happel and I. Reiten, Hereditary categories with tilting object over arbitrary base fields, J. Algebra, in press.
- [6] D. Happel and I. Reiten and S. O. Smalø, Tilting in abelian categories and quasi-tilted algebras, Memoirs Amer. Math. Soc., 575 (1996). Zbl0849.16011
- [7] D. Happel and C. M. Ringel, Tilted algebras, Trans. Amer. Math. Soc. 274 (1982), 399–443. http://dx.doi.org/10.2307/1999116 Zbl0503.16024
- [8] O. Kerner, Stable components of tilted algebras, J. Algebra 162 (1991), 37–57. http://dx.doi.org/10.1016/0021-8693(91)90215-T
- [9] M. Kleiner, A. Skowroński and D. Zacharia, On endomorphism algebras with small homological dimensions, J. Math. Soc. Japan 54 (2002), 621–648. Zbl1035.16007
- [10] H. Lenzing and A. Skowroński, Quasi-tilted algebras of canonical type, Colloq. Math. 71 (1996), 161–181. Zbl0870.16007
- [11] S. Liu, Semi-stable components of an Auslander-Reiten quiver, J. London Math. Soc. 47 (1993), 405–416. Zbl0818.16015
- [12] L. Peng and J. Xiao, On the number of D Tr-orbits containing directing modules, Proc. Amer. Math. Soc. 118 (1993), 753–756. http://dx.doi.org/10.2307/2160117 Zbl0787.16014
- [13] I. Reiten and A. Skowroński, Characterizations of algebras with small homological dimensions, Advances Math., in press. Zbl1051.16011
- [14] I. Reiten and A. Skowroński, Generalized double tilted algebras, J. Math. Soc. Japan, in press. Zbl1071.16011
- [15] C. M. Ringel, “Tame Algebras and Integral Quadratic Forms”, Lecture Notes in Math., Vol. 1099, Springer, 1984. Zbl0546.16013
- [16] D. Simson, Linear Representations of Partially Ordered Sets and Vector Space Categories, Algebra, Logic and Appl. 4 (Gordon and Breach Science Publishers, Amsterdam 1992). Zbl0818.16009
- [17] A. Skowroński, Generalized standard components without oriented cycles, Osaka J. Math. 30 (1993), 515–527. Zbl0818.16017
- [18] A. Skowroński, Generalized standard Auslander-Reiten components, J. Math. Soc. Japan 46 (1994), 517–543. http://dx.doi.org/10.2969/jmsj/04630517 Zbl0828.16011
- [19] A. Skowroński, Regular Auslander-Reiten components containing directing modules, Proc. Amer. Math. Soc. 120 (1994), 19–26. http://dx.doi.org/10.2307/2160162 Zbl0831.16014
- [20] A. Skowroński, Minimal representation-infinite artin algebras, Math. Proc. Camb. Phil. Soc. 116 (1994), 229–243. http://dx.doi.org/10.1017/S0305004100072546 Zbl0822.16010
- [21] A. Skowroński, Directing modules and double tilted algebras, Bull. Polish. Acad. Sci., Ser. Math. 50 (2002), 77–87. Zbl1012.16015
- [22] A. Skowroński, S.O. Smalø and D. Zacharia, On the finiteness of the global dimension of Artin rings, J. Algebra 251 (2002), 475–478. http://dx.doi.org/10.1006/jabr.2001.9130

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