Tensor products of tilting modules

Abstract

We consider whether the tilting properties of a tilting A-module T and a tilting B-module T ^' can convey to their tensor product T ⊗T ^'. The main result is that T ⊗ T ^' turns out to be an (n+ m)-tilting A ⊗ B-module, where T is an m-tilting A-module and T ^' is an n-tilting B-module.

Keywords

Tensor product / tilting module / n-tilting module / endomorphism algebra

References

[1] Anderson F W, Fuller K R. Ring and Categories of Modules. 2nd ed. New York: Springer-Verlag, 1992

[2] Assem I, Simson D, Skowronski A. Elements of the Representation Theory of Associative Algebras I: Techniques of Representation Theory.Cambridge: Cambridge Univ Press, 2006

[3] Bazzoni S. A characterization of n-cotilting and n-tilting modules. J Algebra, 2004, 273(1): 359–372

[4] Brenner S, Butler M C R. Generalizations of the Bernstein-Gelfand-Ponomarev reflection functors. In: Representation Theory II. Lecture Notes in Math, Vol 832. New York: Springer-Verlag, 1980, 103–169

[5] Cartan H, Eilenberg S. Homological Algebra.Princeton: Princeton Univ Press, 1956

[6] Chen M X, Chen Q H. Tensor products of triangular monomial algebras. J Fujian Normal Univ Nat Sci, 2007, 23(6): 19–23

[7] Christopher C G. Tensor products of Young modules. J Algebra, 2012, 366: 12–34

[8] Eilenberg S, Rosenberg A, Zalinsky D. On the dimension of modules and algebras VIII. Nagoya Math J, 1957, 12: 71–93

[9] Happel D. Triangulated Categories in the Representation Theory of Finite Dimensional Algebras. LondonMath Soc Lecture Note Ser, 119. Cambridge: Cambridge Univ Press, 1988

[10] Happel D, Ringel C. Tilted algebras. Trans Amer Math Soc, 1982, 274: 399–443

[11] Hügel A L, Coelho F U. Infinitely generated tilting modules of finite projective dimension. Forum Math, 2001, 13: 239–250

[12] Miyashita Y. Tilting modules of finite projective dimension. Math Z, 1986, 193: 113–146

[13] Rickard J. Morita theory for derived categories. J Lond Math Soc, 1989, 39: 436–456

[14] Rotman J J. An Introduction to Homological Algebra. 2nd ed.New York: Springer, 2009

[15] Yang J W. Tensor products of strongly graded vertex algebras and their modules. J Pure Appl Algebra, 2013, 217(2): 348–363

[16] Zhou B X. On the tensor product of left modules and their homological dimensions. J Math Res Exposition, 1981, 1: 17–24

[17] Zhou B X. Homological Algebra.Beijing: Science Press, 1988 (in Chinese)

RIGHTS & PERMISSIONS

2016 Higher Education Press and Springer-Verlag Berlin Heidelberg