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 is a tilting module and , then T(A) is tame if and only if T(B) is tame.
@article{bwmeta1.element.bwnjournal-article-fmv149i2p171bwm, author = {Ibrahim Assem and Jos\'e de la Pe\~na}, title = {On the tameness of trivial extension algebras}, journal = {Fundamenta Mathematicae}, volume = {149}, year = {1996}, pages = {171-181}, zbl = {0851.16013}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-fmv149i2p171bwm} }
Assem, Ibrahim; de la Peña, José. On the tameness of trivial extension algebras. Fundamenta Mathematicae, Tome 149 (1996) pp. 171-181. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv149i2p171bwm/
[00000] [1] I. Assem, Tilting theory - an introduction, in: Topics in Algebra, Banach Center Publ. 26, Part 1, PWN, Warszawa, 1990, 127-180.
[00001] [2] I. Assem, D. Happel and O. Roldán, Representation-finite trivial extension algebras, J. Pure Appl. Algebra 33 (1984), 235-242. | Zbl 0564.16027
[00002] [3] I. Assem, J. Nehring and A. Skowroński, Domestic trivial extension of simply connected algebras, Tsukuba J. Math. 13 (1989), 31-72. | Zbl 0686.16011
[00003] [4] M. Auslander and I. Reiten, Representation theory of artin algebras III, IV, Comm. Algebra 3 (1975), 239-294 and 5 (1977), 443-518.
[00004] [5] Yu. A. Drozd, Tame and wild matrix problems, in: Representation Theory II, Lecture Notes in Math. 832, Springer, Berlin, 1980, 240-258.
[00005] [6] R. M. Fossum, P. A. Griffith and I. Reiten, Trivial Extensions of Abelian Categories, Lecture Notes in Math. 456, Springer, Berlin, 1975. | Zbl 0303.18006
[00006] [7] D. Happel and C. M. Ringel, Tilted algebras, Trans. Amer. Math. Soc. 274 (2) (1982), 399-443. | Zbl 0503.16024
[00007] [8] D. Hughes and J. Waschbüsch, Trivial extensions of tilted algebras, Proc. London Math. Soc. (3) 46 (1983), 347-364. | Zbl 0488.16021
[00008] [9] R. Martínez-Villa, Properties that are left invariant under stable equivalence, Comm. Algebra 18 (12) (1990), 4141-4169. | Zbl 0724.16002
[00009] [10] J. Nehring, Trywialne rozszerzenia wielomianowego wzrostu [Trivial extensions of polynomial growth], Ph.D. thesis, Nicholas Copernicus Univ., 1989 (in Polish).
[00010] [11] J. Nehring, Polynomial growth trivial extensions of non-simply connected algebras, Bull. Polish Acad. Sci. Math. 36 (1988), 441-445. | Zbl 0777.16008
[00011] [12] J. Nehring and A. Skowroński, Polynomial growth trivial extensions of simply connected algebras, Fund. Math. 132 (1989), 117-134. | Zbl 0677.16008
[00012] [13] J. A. de la Peña, Functors preserving tameness, ibid. 137 (1991), 177-185. | Zbl 0790.16014
[00013] [14] J. A. de la Peña, Constructible functors and the notion of tameness, Comm. Algebra, to appear. | Zbl 0858.16007
[00014] [15] C. M. Ringel, Tame Algebras and Integral Quadratic Forms, Lecture Notes in Math. 1099, Springer, Berlin, 1984.
[00015] [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.
[00016] [17] H. Tachikawa and T. Wakamatsu, Tilting functors and stable equivalences for selfinjective algebras, J. Algebra 109 (1987), 138-165. | Zbl 0616.16012