Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-15T13:06:37.628Z Has data issue: false hasContentIssue false
  • English
  • Français

Complete slices and homological properties of tilted algebras

Published online by Cambridge University Press:  18 May 2009

Ibrahim Assem  and
Flávio Ulhoa Coelho
Ibrahim Assem
Affiliation:
Mathématiques et Informatique, Université de Sherbrooke, Sherbrooke, Quebec J1K 2R1, Canada E-mail address: iassem@vml.si.usherb.ca
Flávio Ulhoa Coelho
Affiliation:
Departamento de Matemática-IME, Universidade de Sāo Paulo, CP 20570, São Paulo, SP, 01452-990, Brasil E-mail address: fucoelho@ime.usp.br
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

It is reasonable to expect that the representation theory of an algebra (finite dimensional over a field, basic and connected) can be used to study its homological properties. In particular, much is known about the structure of the Auslander-Reiten quiver of an algebra, which records most of the information we have on its module category. We ask whether one can predict the homological dimensions of a module from its position in the Auslander-Reiten quiver. We are particularly interested in the case where the algebra is a tilted algebra. This class of algebras of global dimension two, introduced by Happel and Ringel in [7], has since then been the subject of many investigations, and its representation theory is well understood by now (see, for instance, [1], [7], [8], [9], [11], [13]).In this case, the most striking feature of the Auslander-Reiten quiver is the existence of complete slices, which reproduce the quiver of the hereditary algebra from which the tilted algebra arises. It follows from well-known results that any indecomposable successor (or predecessor) of a complete slice has injective (or projective, respectively) dimension at most one, from which one deduces that a tilted algebra is representation-finite if and only if both the projective and the injective dimensions of almost all (that is, all but at most finitely many non-isomorphic) indecomposable modules equal two (see (3.1) and (3.2)). On the other hand, the authors have shown in [2, (3.4)] that a representation-infinite algebra is concealed if and only if both the projective and the injective dimensions of almost all indecomposable modules equal one (see also [14]). This leads us to consider, for tilted algebras which are not concealed, the case when the projective (or injective) dimension of almost all indecomposable successors (or predecessors, respectively) of a complete slice equal two. In order to answer this question, we define the notions of left and right type of a tilted algebra, then those of reduced left and right types (see (2.2) and (3.4) for the definitions).

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 36 , Issue 3 , September 1994 , pp. 347 - 354
Copyright
Copyright © Glasgow Mathematical Journal Trust 1994

References

REFERENCES

1.Assem, I., Tilting theory – an introduction, in Topics in algebra, Banach Centre Publications 26 (PWN, 1990), 127–180.Google Scholar
2.Assem, I. and Coelho, F. U., Glueings of tilted algebras, J. Pure Appl. Algebra, to appear.Google Scholar
3.Auslander, M. and Reiten, I., Representation theory of Artin algebras. III, almost split sequences, and IV, invariants given by almost split sequences, Comm. Algebra 3 (1975), 239–294 and 5 (1977), 443–518.CrossRefGoogle Scholar
4.Auslander, M. and Smalø, S. O., Preprojective modules over Artin algebras, J. Algebra 66 (1980), 61–122.CrossRefGoogle Scholar
5.Bautista, R., Sections in Auslander-Reiten quivers, Proc. ICRA II (Ottawa, 1979), Lecture Notes in Mathematics 832 (Springer, 1980), 74–96.Google Scholar
6.Gabriel, P. and Rojter, A. V., Representations of finite-dimensional algebras, Encyclopaedia of Mathematical Sciences 73, Algebra VIII (Springer, 1992).Google Scholar
7.Happel, D. and Ringel, C. M., Tilted algebras, Trans. Amer. Math. Soc. 274 (1982), 399–443.CrossRefGoogle Scholar
8.Kerner, O., Tilting wild algebras, J. London Math. Soc. (2) 39 (1989), 29–47.CrossRefGoogle Scholar
9.Liu, S., The connected components of the Auslander-Reiten quiver of a tilted algebra, J. Algebra 161 (1993), 505–523.CrossRefGoogle Scholar
10.Liu, S., Semi-stable components of an Auslander-Reiten quiver, J. London Math. Soc. (2) 47 (1993), 405–416.CrossRefGoogle Scholar
11.Ringel, C. M., Tame algebras and integral quadratic forms, Lecture Notes in Mathematics 1099 (Springer, 1984).CrossRefGoogle Scholar
12.Ringel, C. M., Representation theory of finite-dimensional algebras, in Representations of algebras (Durham, 1985), London Math. Soc. Lecture Notes Series 116 (Cambridge University Press, 1986) 7–79.CrossRefGoogle Scholar
13.Ringel, C. M., The regular components of the Auslander-Reiten quiver of a tilted algebra, Chinese Ann. Math. Ser. B 9 (1988), 1–18.Google Scholar
14.Skowroński, A., Minimal representation-infinite artin algebras, Math. Proc. Cambridge Philos. Soc, to appear.Google Scholar