Skip to main content Accessibility help


  • JIAQUN WEI (a1) (a2)


Let $R$ be a ring and $T$ be a good Wakamatsu-tilting module with $S=\text{End}(T_{R})^{op}$ . We prove that $T$ induces an equivalence between stable repetitive categories of $R$ and $S$ (i.e., stable module categories of repetitive algebras $\hat{R}$ and ${\hat{S}}$ ). This shows that good Wakamatsu-tilting modules seem to behave in Morita theory of stable repetitive categories as that tilting modules of finite projective dimension behave in Morita theory of derived categories.



Hide All

The author is supported by the Natural Science Foundation of China (Grant No. 11771212) and a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.



Hide All
[1]Angeleri-Hügel, L. and Coelho, F.U., Infinitely generated tilting modules of finite projective dimension, Forum Math. 13 (2001), 239250.
[2]Asashiba, H., A covering technique for derived equivalence, J. Algebra 191 (1997), 382415.
[3]Auslander, M. and Reiten, I., Applications of contravariantly finite subcategories, Adv. Math. 86 (1991), 111152.
[4]Bazzoni, S., A characterization of n-cotilting and n-tilting modules, J. Algebra 273(1) (2004), 359372.
[5]Bongartz, K., “Tilted algebras”, in Representations of Algebras, Lecture Notes in Mathematics 903, Springer-Verlag, Berlin/Heidelberg/New York, 1981, 2638.
[6]Brenner, S. and Butler, M. C. R., Generalizations of the Bernstein–Gelfand–Ponomarev Reflection Functors, Lecture Notes in Mathematics 832, 1980, 103170.
[7]Chen, Q., Derived equivalence of repetitive algebras, Adv. Math. (Chinese) 37(2) (2008), 189196.
[8]Chen, X., The singularity category of an algebra with radical square zero, Doc. Math. 16 (2011), 921936.
[9]Chen, X. and Wei, J., Wakamatsu’s equivalence revisited, arXiv:1610.09649.
[10]Cline, E., Parshall, B. and Scott, L., Derived categories and Morita theory, J. Algebra 104 (1986), 397409.
[11]Colby, R. and Fuller, K. R., Tilting and torsion theory counter equivalences, Comm. Algebra 23(13) (1995), 48334849.
[12]Colby, R. and Fuller, K. R., Tilting, cotilting and serially tilted rings, Comm. Algebra 25(10) (1997), 32253237.
[13]Colpi, R., D’Este, G. and Tonolo, A., Quasi-tilting modules and counter equivalences, J. Algebra 191 (1997), 461494.
[14]Colpi, R. and Trlifaj, J., Tilting modules and tilting torsion theories, J. Algebra 178 (1995), 614634.
[15]Enochs, E. E. and Jenda, O. M. G., Relative Homological Algebra, De Gruyter Expositions in Mathematics 30, Walter De Gruyter, Berlin/New York, 2000.
[16]Göbel, R. and Trlifaj, J., Approximations and Endomorphism Algebras of Modules, De Gruyter Expositions in Mathematics 41, Walter de Gruyter, Berlin/New York, 2012.
[17]Green, E. L., Reiten, I. and Solberg, Ø., Dualities on Generalized Koszul Algebras, Mem. Amer. Math. Soc. 159 (2002), xvi+67 pp.
[18]Happel, D., Triangulated Categories in the Representation Theory of Finite Dimension Algebras, London Mathematical Society Lecture Notes Series 119, Cambridge University Press, Cambridge, 1988.
[19]Happel, D. and Ringel, C. M., Tilted algebras, Trans. Amer. Math. Soc. 174 (1982), 399443.
[20]Hughes, D. and Waschbüsch, J., Trivial extensions of tilted algebras, Proc. Lond. Math. Soc. 46 (1983), 347364.
[21]Keller, B., Deriving DG categories, Ann. Sci. Éc. Norm. Supér. 27 (1994), 63102.
[22]Mantese, F. and Reiten, I., Wakamatsu Tilting modules, J. Algebra 278 (2004), 532552.
[23]Miyashita, Y., Tilting modules of finite projective dimension, Math. Z. 193 (1986), 113146.
[24]Rickard, J., Morita theory for derived categories, J. Lond. Math. Soc. 39 (1989), 436456.
[25]Wakamatsu, T., On modules with trivial self-extensions, J. Algebra 114 (1988), 106114.
[26]Wakamatsu, T., Stable equivalence for self-injective algebras and a generalization of tilting modules, J. Algebra 134 (1990), 298325.
[27]Wakamatsu, T., On constructing stable equivalent functor, J. Algebra 148 (1992), 277288.
[28]Yamaura, K., Realizing stable categories as derived categories, Adv. Math. 248 (2013), 784819.
[29]Yoshino, Y., “Modules of G-dimension zero over local rings with the cube of maximal ideal being zero”, in Commutative Algebra, Singularities and Computer Algebra, NATO Sci. Ser. I Math. Phys. Chem. 115, Kluwer, Dordrecht, 2003, 255273.
MathJax is a JavaScript display engine for mathematics. For more information see

MSC classification


  • JIAQUN WEI (a1) (a2)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed