On the tameness of trivial extension algebras

Ibrahim Assem; José de la Peña

Fundamenta Mathematicae (1996)

  • Volume: 149, Issue: 2, page 171-181
  • ISSN: 0016-2736

Abstract

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For a finite dimensional algebra A over an algebraically closed field, let T(A) denote the trivial extension of A by its minimal injective cogenerator bimodule. We prove that, if T A is a tilting module and B = E n d T A , then T(A) is tame if and only if T(B) is tame.

How to cite

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Assem, Ibrahim, and de la Peña, José. "On the tameness of trivial extension algebras." Fundamenta Mathematicae 149.2 (1996): 171-181. <http://eudml.org/doc/212115>.

@article{Assem1996,
abstract = {For a finite dimensional algebra A over an algebraically closed field, let T(A) denote the trivial extension of A by its minimal injective cogenerator bimodule. We prove that, if $T_A$ is a tilting module and $B=End T_A$, then T(A) is tame if and only if T(B) is tame.},
author = {Assem, Ibrahim, de la Peña, José},
journal = {Fundamenta Mathematicae},
keywords = {stable equivalence; tame algebras; finite dimensional algebras; trivial extensions; minimal injective cogenerators; tilting modules},
language = {eng},
number = {2},
pages = {171-181},
title = {On the tameness of trivial extension algebras},
url = {http://eudml.org/doc/212115},
volume = {149},
year = {1996},
}

TY - JOUR
AU - Assem, Ibrahim
AU - de la Peña, José
TI - On the tameness of trivial extension algebras
JO - Fundamenta Mathematicae
PY - 1996
VL - 149
IS - 2
SP - 171
EP - 181
AB - For a finite dimensional algebra A over an algebraically closed field, let T(A) denote the trivial extension of A by its minimal injective cogenerator bimodule. We prove that, if $T_A$ is a tilting module and $B=End T_A$, then T(A) is tame if and only if T(B) is tame.
LA - eng
KW - stable equivalence; tame algebras; finite dimensional algebras; trivial extensions; minimal injective cogenerators; tilting modules
UR - http://eudml.org/doc/212115
ER -

References

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  1. [1] I. Assem, Tilting theory - an introduction, in: Topics in Algebra, Banach Center Publ. 26, Part 1, PWN, Warszawa, 1990, 127-180. 
  2. [2] I. Assem, D. Happel and O. Roldán, Representation-finite trivial extension algebras, J. Pure Appl. Algebra 33 (1984), 235-242. Zbl0564.16027
  3. [3] I. Assem, J. Nehring and A. Skowroński, Domestic trivial extension of simply connected algebras, Tsukuba J. Math. 13 (1989), 31-72. Zbl0686.16011
  4. [4] M. Auslander and I. Reiten, Representation theory of artin algebras III, IV, Comm. Algebra 3 (1975), 239-294 and 5 (1977), 443-518. 
  5. [5] Yu. A. Drozd, Tame and wild matrix problems, in: Representation Theory II, Lecture Notes in Math. 832, Springer, Berlin, 1980, 240-258. 
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  7. [7] D. Happel and C. M. Ringel, Tilted algebras, Trans. Amer. Math. Soc. 274 (2) (1982), 399-443. Zbl0503.16024
  8. [8] D. Hughes and J. Waschbüsch, Trivial extensions of tilted algebras, Proc. London Math. Soc. (3) 46 (1983), 347-364. Zbl0488.16021
  9. [9] R. Martínez-Villa, Properties that are left invariant under stable equivalence, Comm. Algebra 18 (12) (1990), 4141-4169. Zbl0724.16002
  10. [10] J. Nehring, Trywialne rozszerzenia wielomianowego wzrostu [Trivial extensions of polynomial growth], Ph.D. thesis, Nicholas Copernicus Univ., 1989 (in Polish). 
  11. [11] J. Nehring, Polynomial growth trivial extensions of non-simply connected algebras, Bull. Polish Acad. Sci. Math. 36 (1988), 441-445. Zbl0777.16008
  12. [12] J. Nehring and A. Skowroński, Polynomial growth trivial extensions of simply connected algebras, Fund. Math. 132 (1989), 117-134. Zbl0677.16008
  13. [13] J. A. de la Peña, Functors preserving tameness, ibid. 137 (1991), 177-185. Zbl0790.16014
  14. [14] J. A. de la Peña, Constructible functors and the notion of tameness, Comm. Algebra, to appear. Zbl0858.16007
  15. [15] C. M. Ringel, Tame Algebras and Integral Quadratic Forms, Lecture Notes in Math. 1099, Springer, Berlin, 1984. 
  16. [16] H. Tachikawa, Selfinjective algebras and tilting theory, in: Representation Theory I. Finite Dimensional Algebras, Lectures Notes in Math. 1177, Springer, Berlin, 1986, 272-307. 
  17. [17] H. Tachikawa and T. Wakamatsu, Tilting functors and stable equivalences for selfinjective algebras, J. Algebra 109 (1987), 138-165. Zbl0616.16012

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