Tilted algebras

Happel, Dieter; Ringel, Claus Michael

doi:10.1090/S0002-9947-1982-0675063-2

Tilted algebras
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Trans. Amer. Math. Soc. 274 (1982), 399-443 Request permission

Abstract:

Let $A$ be a finite dimensional hereditary algebra over a field, with $n$ simple $A$-modules. An $A$-module $T_A$ with $n$ pairwise nonisomorphic indecomposable direct summands and satisfying ${\text {Ex}}{{\text {t}}^1}({T_A}, {T_A}) = 0$ is called a tilting module, and its endomorphism ring $B$ is a tilted algebra. A tilting module defines a (usually nonhereditary) torsion theory, and the indecomposable $B$-modules are in one-to-one correspondence to the indecomposable $A$-modules which are either torsion or torsionfree. One of the main results of the paper asserts that an algebra of finite representation type with an indecomposable sincere representation is a tilted algebra provided its Auslander-Reiten quiver has no oriented cycles. In fact, tilting modules are introduced and studied for any finite dimensional algebra, generalizing recent results of Brenner and Butler.

References

Maurice Auslander and Idun Reiten, Representation theory of Artin algebras. IV. Invariants given by almost split sequences, Comm. Algebra 5 (1977), no. 5, 443–518. MR 439881, DOI 10.1080/00927877708822180
M. Auslander and Sverre O. Smalø, Almost split sequences in subcategories, J. Algebra 69 (1981), no. 2, 426–454. MR 617088, DOI 10.1016/0021-8693(81)90214-3
R. Bautista, Sections in Auslander-Reiten quivers, Representation theory, II (Proc. Second Internat. Conf., Carleton Univ., Ottawa, Ont., 1979) Lecture Notes in Math., vol. 832, Springer, Berlin-New York, 1980, pp. 74–96. MR 607149

Sections in Auslander-Reiten components

Auslander-Reiten quivers for certain algebras of finite representation type

Sheila Brenner and M. C. R. Butler, Generalizations of the Bernstein-Gel′fand-Ponomarev reflection functors, Representation theory, II (Proc. Second Internat. Conf., Carleton Univ., Ottawa, Ont., 1979) Lecture Notes in Math., vol. 832, Springer, Berlin-New York, 1980, pp. 103–169. MR 607151
Vlastimil Dlab and Claus Michael Ringel, Indecomposable representations of graphs and algebras, Mem. Amer. Math. Soc. 6 (1976), no. 173, v+57. MR 447344, DOI 10.1090/memo/0173
Peter Gabriel, Auslander-Reiten sequences and representation-finite algebras, Representation theory, I (Proc. Workshop, Carleton Univ., Ottawa, Ont., 1979), Lecture Notes in Math., vol. 831, Springer, Berlin, 1980, pp. 1–71. MR 607140

Zur Darstellungstheorie von Köchern endlichen und zahmen Typs. Relative Invarianten und Subgenerische Orbiten

Manabu Harada and Youshin Sai, On categories of indecomposable modules. I, Osaka Math. J. 7 (1970), 323–344. MR 286859
C. Riedtmann, Algebren, Darstellungsköcher, Überlagerungen und zurück, Comment. Math. Helv. 55 (1980), no. 2, 199–224 (German). MR 576602, DOI 10.1007/BF02566682
Claus Michael Ringel, The rational invariants of the tame quivers, Invent. Math. 58 (1980), no. 3, 217–239. MR 571574, DOI 10.1007/BF01390253

Tame algebras

Report on the Brauer-Thrall conjecures: Rojter’s theorem and the theorem of Nazarova-Rojter

Applications of morphisms determined by models. Representation theory of algebras

Maurice Auslander, María Inés Platzeck, and Idun Reiten, Coxeter functors without diagrams, Trans. Amer. Math. Soc. 250 (1979), 1–46. MR 530043, DOI 10.1090/S0002-9947-1979-0530043-2
Klaus Bongartz and Claus Michael Ringel, Representation-finite tree algebras, Representations of algebras (Puebla, 1980) Lecture Notes in Math., vol. 903, Springer, Berlin-New York, 1981, pp. 39–54. MR 654702
Dieter Happel and Claus Michael Ringel, Construction of tilted algebras, Representations of algebras (Puebla, 1980) Lecture Notes in Math., vol. 903, Springer, Berlin-New York, 1981, pp. 125–144. MR 654707