Abstract
For classes of algebras defined in nonhomological terms it is often of interest to look for a homological description. For example the Auslander algebras, that is the algebras which are obtained as endomorphism rings of a direct sum of representatives from each isomorphism class of the indecomposable modules over a non semisimple algebra of finite representation type, are characterized by having global dimension and dominant dimension equal to 2. Another important class of artin algebras is the class of tilted algebras, and it is well known that they have global dimension at most 2. In addition, they have no indecomposable module having both projective and injective dimension equal to 2. However these properties do not characterize tilted algebras. But it turns out that they characterize a natural generalization of tilted algebras obtained by introducing the notion of tilting objects in hereditary abelian categories more general than module categories over hereditary algebras. The endomorphism algebras of these tilting objects are called quasitilted algebras. And this is precisely the class which is characterized by the homological properties above. Besides the tilted algebras, also the canonical algebras [Ri] belong to this class.
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References
Auslander, M. and Reiten, I., Applications of Contravariantly finite Subcategories, Adv. in Math.,Vol. 86,No 1 (1991), 111–152.
Auslander, M. and Smalo, S. O. Preprojective modules over actin algebras, J. Algebra 66 (1980) 61–122.
Auslander, M. and Smalo, S. O., Almost split sequences in subcategories, J. Algebra 69 (1981), 426–454; Addendum J. Algebra 71 (1981), 592–594.
Baer, D., Wild hereditary arlin algebras and linear methods, Manuscripta Math. 55 (1986), 69–82.
Beilinson, A. A., Coherent Sheaves on IP and problems of linear algebra, Func. Anal. and Appl. 12 (1978) 212–214.
Bautista, R. and Larrion, F., Auslander-Reiten Quivers for Certain Algebras of Finite Representation Type, J. London Math. Soc. (2), 26 (1982), 43–52.
Geigle, W. and Lenzing, H., Perpendicular categories with applications to representations and sheaves, J. Algebra 144 (1991), 273–343.
Happel, D., Triangulated Categories in the Representation Theory of Finite Dimensional Algebras, London Math. Soc. Lecture Note Series, 119, 1988.
Happel, D. and Ringel, C. M., Tilted algebras, Trans. Amer. M.th. Soc., 274 (1982), 399–443.
Happel, D., Reiten, I. and Smalo, S. O., Tilting in Abelian Categories and Quasitilted Algebras,in preparation.
Hartshorne, R., Algebraic Geometry, Springer, Heidelberg 1977.
Igusa, K., Platzeck, M. I., Todorov, G. and Zacharia, D., Auslander Algebras of Finite Representation Type, Comm. in Algebra, 15(1&2), 377–424 (1987).
Kerner, O., Elementary Stones, Preprint Düsseldorf 1992.
Kerner, O. and Lukas, F., Elementary modules, Preprint Düsseldorf 1992.
Lukas, F., Elementare Moduln liber wilden erblichen Algebren, Dissertation, Düsseldorf 1992.
Miyashita, T., Tilting modules of finite injective dimension,Math. Zeit. 193(1986),113146.
Rickard, J., Morita theory for Derived Categories,Journal of the London Math. Soc., 39, 1989 (436–456).
Ringel, C. M.; Tame Algebras and Integral Quadratic Forms, Lecture Notes in Mathematics 1099, Springer, Berlin, 1984.
Reiten, I., Skowronski, A. and Smalo, S. O., Short chains and short cycles of modules, Proc. Amer. Math. Soc. (to appear)
Verdier, J.-L., Catégories dérivées, étal 0, SGA 4 1/2, Lecture Notes in Mathematics 569, (262–311) Berlin, 1977.
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Happel, D., Reiten, I., Smalø, S.O. (1994). Quasitilted Algebras. In: Dlab, V., Scott, L.L. (eds) Finite Dimensional Algebras and Related Topics. NATO ASI Series, vol 424. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-1556-0_8
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DOI: https://doi.org/10.1007/978-94-017-1556-0_8
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