# Finiteness of the strong global dimension of radical square zero algebras

Otto Kerner; Andrzej Skowroński; Kunio Yamagata; Dan Zacharia

Open Mathematics (2004)

- Volume: 2, Issue: 1, page 103-111
- ISSN: 2391-5455

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topOtto Kerner, et al. "Finiteness of the strong global dimension of radical square zero algebras." Open Mathematics 2.1 (2004): 103-111. <http://eudml.org/doc/268840>.

@article{OttoKerner2004,

abstract = {The strong global dimension of a finite dimensional algebra A is the maximum of the width of indecomposable bounded differential complexes of finite dimensional projective A-modules. We prove that the strong global dimension of a finite dimensional radical square zero algebra A over an algebraically closed field is finite if and only if A is piecewise hereditary. Moreover, we discuss results concerning the finiteness of the strong global dimension of algebras and the related problem on the density of the push-down functors associated to the canonical Galois coverings of the trivial extensions of algebras by their repetitive algebras.},

author = {Otto Kerner, Andrzej Skowroński, Kunio Yamagata, Dan Zacharia},

journal = {Open Mathematics},

keywords = {Primary 16D50; 16E10; 18E30; Secondary 16G10},

language = {eng},

number = {1},

pages = {103-111},

title = {Finiteness of the strong global dimension of radical square zero algebras},

url = {http://eudml.org/doc/268840},

volume = {2},

year = {2004},

}

TY - JOUR

AU - Otto Kerner

AU - Andrzej Skowroński

AU - Kunio Yamagata

AU - Dan Zacharia

TI - Finiteness of the strong global dimension of radical square zero algebras

JO - Open Mathematics

PY - 2004

VL - 2

IS - 1

SP - 103

EP - 111

AB - The strong global dimension of a finite dimensional algebra A is the maximum of the width of indecomposable bounded differential complexes of finite dimensional projective A-modules. We prove that the strong global dimension of a finite dimensional radical square zero algebra A over an algebraically closed field is finite if and only if A is piecewise hereditary. Moreover, we discuss results concerning the finiteness of the strong global dimension of algebras and the related problem on the density of the push-down functors associated to the canonical Galois coverings of the trivial extensions of algebras by their repetitive algebras.

LA - eng

KW - Primary 16D50; 16E10; 18E30; Secondary 16G10

UR - http://eudml.org/doc/268840

ER -

## References

top- [1] I. Assem, J. Nehring and A. Skowroński, “Domestic trivial extensions of simply connected algebras”, Tsukuba J. Math., Vol. 13, (1989), pp. 31–72. Zbl0686.16011
- [2] I. Assem and A. Skowroński, “Algebras with cycle-finite derived categories”, Math. Ann., Vol. 280, (1988), pp. 441–463. http://dx.doi.org/10.1007/BF01456336 Zbl0617.16017
- [3] I. Assem and A. Skowroński, “On tame repetitive algebras”, Fund. Math., Vol. 142, (1993), pp. 59–84. Zbl0833.16010
- [4] K. Bongartz and P. Gabriel, “Covering spaces in representations theory”, Invent. Math., Vol. 65, (1982), pp. 331–378. http://dx.doi.org/10.1007/BF01396624 Zbl0482.16026
- [5] P. Dowbor and A. Skowroński, “On Galois coverings of tame algebras”, Arch. Math., Vol. 44, (1985), pp. 522–529. http://dx.doi.org/10.1007/BF01193992 Zbl0576.16029
- [6] P. Dowbor and A. Skowroński, “Galois coverings of representation-infinite algebras”, Comment. Math. Helv., Vol. 62, (1987), pp. 311–337. Zbl0628.16019
- [7] K. Erdmann, O. Kerner and A. Skowroński, “Self-injective algebras of wild tilted type”, J. Pure Appl. Algebra, Vol. 149, (2000), pp. 127–176. http://dx.doi.org/10.1016/S0022-4049(00)00035-9 Zbl0994.16015
- [8] P. Gabriel, “The universal cover of a representation-finite algebra” In: Representations of Algebras, Lecture Notes in Math., Vol. 903, Springer, 1981, pp. 68–105.
- [9] E.L. Green, “Graphs with relations, coverings and group graded algebras”, Trans. Amer. Math. Soc., Vol. 279, (1983), pp. 297–310. http://dx.doi.org/10.2307/1999386 Zbl0536.16001
- [10] D. Happel, “On the derived category of a finite dimensional algebra”, Comment. Math. Helv, Vol. 62, (1987), pp. 339–389. Zbl0626.16008
- [11] D. Happel, “A characterization of hereditary categories with tilting object”, Invent. Math., Vol. 144, (2001), pp. 381–398. http://dx.doi.org/10.1007/s002220100135 Zbl1015.18006
- [12] D. Happel, I. Reiten and S.O. Smalø, Tilting in abelian categories and quasitilted algebras, Mem. Amer. Math. Soc., Providence, Rhode Island, 1996. Zbl0849.16011
- [13] D. Happel, I. Reiten and S.O. Smalø, “Piecewise hereditary algebras”, Arch. Math., Vol. 66, (1996), pp. 182–186. http://dx.doi.org/10.1007/BF01195702 Zbl0865.16009
- [14] D. Happel, J. Rickard and A. Schofield, “Piecewise hereditary algebras”, Bull. London Math. Soc., Vol. 20, (1988), pp. 23–28. Zbl0638.16018
- [15] D. Hughes and J. Waschbüsch, “Trivial extensions of tilted algebras”, Proc. London Math. Soc., Vol. 46, (1983), pp. 347–364. Zbl0488.16021
- [16] H. Lenzing and A. Skowroński, “Selfinjective algebras of wild canonical type”, Colloq. Math., Vol. 96, (2003), pp. 245–275. http://dx.doi.org/10.4064/cm96-2-9 Zbl1050.16009
- [17] J. Nehring and A. Skowroński, “Polynomial growth trivial extensions of simply connected algebras”, Fund. Math., Vol. 132, (1989), pp. 117–134. Zbl0677.16008
- [18] C.M. Ringel, Tame algebras and integral guadratic forms, Lecture Notes in Math., Springer, Berlin, 1984.
- [19] J. Schröer, “On the quiver with relations of a repetitive algebra”, Arch. Math., Vol. 72, (1999), pp. 426–432. http://dx.doi.org/10.1007/s000130050351 Zbl0937.16018
- [20] A. Skowroński, “Generalization of Yamagata’s theorem on trivial extensions”, Arch. Math., Vol. 48, (1987), pp. 68–76. http://dx.doi.org/10.1007/BF01196357 Zbl0634.16013
- [21] A. Skowroński, “On algebras with finite strong global dimension”, Bull. Polish. Acad. Sci., Vol. 35, (1987), pp. 539–547. Zbl0642.16023
- [22] A. Skowroński, “Tame quasi-tilted algebras”, J. Algebra, Vol. 203, (1998), pp. 470–490. http://dx.doi.org/10.1006/jabr.1997.7328
- [23] K. Yamagata, “On algebras whose trivial extensions are of finite representation type”, In: Representations of Algebras, Lecture Notes in Math., Vol. 903, Springer, 1981, pp. 364–371.

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