manuscripta mathematica

, Volume 57, Issue 1, pp 1–31 | Cite as

Lower bounds for faithful, preinjective modules

  • Luise Unger


Let\(k\overrightarrow \Delta \) be the path algebra for some representation-infinite quiver\(\overrightarrow \Delta \) over some field k. There exists a bound\(m(\overrightarrow \Delta )\) such that τmI is faithful for all indecomposable injective\(k\overrightarrow \Delta \)-modules I and all\(m \geqslant m(\overrightarrow {\Delta )} \), and such that there → exists an indecomposable injective\(k\overrightarrow \Delta \)-module J satisfying that\({}_\tau m(\overrightarrow {\Delta )} - I_J \) J is not faithful, τ denotes the Auslander-Reiten-translation. Let m(Δ) be the maximum of the\(m(\overrightarrow {\Delta )} \) taken over all possible orientations of the underlying graph Δ. In this article we determine the bounds m(Δ) for representation-infinite quivers\(\overrightarrow \Delta \) for which Δ is a tree.


Number Theory Algebraic Geometry Topological Group Path Algebra Preinjective Module 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    R. Bautista, S. Brenner: On the number of terms in the middle of an almost split sequence and related results, Representations of algebras, SLM 903, (1980), 1–8Google Scholar
  2. [2]
    R. Bautista, S. Brenner: Replication numbers for non-Dynkin sectional subgraphs in finite Auslander-Reiten-quivers and some properties of Weyl roots, Proc. London Math. Soc. (3), 47, (1983), 429–462Google Scholar
  3. [3]
    I.N. Bernstein, I.M. Gelfand, V.A. Ponomarev: Coxeter functors and Gabriels's theorem, Uspechi Mat. Nauk. 28, (1973), English translation: Russian Math. Surveys, 28, (1973), 17–33Google Scholar
  4. [4]
    K. Bongartz, P. Gabriel: Covering spaces in representation theory, Invent, math. 65, (1982), 331–378Google Scholar
  5. [5]
    V. Dlab, C.M. Ringel: Indecomposable representations of graphs and algebras, Memoirs Amer. Math. Soc., 173, (1972)Google Scholar
  6. [6]
    D. Happel, C.M. Ringel: Tilted algebras, Trans. Amer. Math. Soc. 274, (1982), 399–443Google Scholar
  7. [7]
    D. Happel, C.M. Ringel: Construction of tilted algebras, Representations of algebras, SLM 903, (1980), 125–143Google Scholar
  8. [8]
    C. Riedtmann: Algebren, Darstellungsköcher, Überlagerungen und zurück, Com. Math. Helv. 55, (1980), 199–224Google Scholar
  9. [9]
    C.M. Ringel: Separating tubular series, Sém. d'algèbre P. Dubreil et M.P. Malliavin (1982), SLM 1029, (1983), 134–158Google Scholar
  10. [10]
    C.M. Ringel: Tame algebras and integral quadratic forms, SLM 1099, 1984Google Scholar
  11. [11]
    L. Unger: Preinjective components of trees, to appear in SLMGoogle Scholar

Copyright information

© Springer-Verlag 1986

Authors and Affiliations

  • Luise Unger
    • 1
  1. 1.Fakultät für MathematikUniversität BielefeldBielefeldFRG

Personalised recommendations