Tensor products of tilting modules. (English) Zbl 1379.16007
Summary: We consider whether the tilting properties of a tilting \(A\)-module \(T\) and a tilting \(B\)-module \(T^\prime\) can convey to their tensor product \(T\otimes T^\prime\): The main result is that \(T\otimes T^\prime\) turns out to be an \((n + m)\)-tilting \(A\otimes B\)-module, where T is an \(m\)-tilting \(A\)-module and \(T^\prime\) is an \(n\)-tilting \(B\)-module.
MSC:
| 16G10 | Representations of associative Artinian rings |
| 16G20 | Representations of quivers and partially ordered sets |
References:
| [1] | Anderson F W, Fuller K R. Ring and Categories of Modules. 2nd ed. New York: Springer-Verlag, 1992 · Zbl 0765.16001 · doi:10.1007/978-1-4612-4418-9 |
| [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 · Zbl 1092.16001 · doi:10.1017/CBO9780511614309 |
| [3] | Bazzoni S. A characterization of n-cotilting and n-tilting modules. J Algebra, 2004, 273(1): 359-372 · Zbl 1051.16007 · doi:10.1016/S0021-8693(03)00432-0 |
| [4] | Brenner, S.; Butler, M. C R., Generalizations of the Bernstein-Gelfand-Ponomarev re ection functors, 103-169 (1980) · Zbl 0446.16031 |
| [5] | Cartan H, Eilenberg S. Homological Algebra. Princeton: Princeton Univ Press, 1956 · Zbl 0075.24305 |
| [6] | Chen M X, Chen Q H. Tensor products of triangular monomial algebras. J Fujian Normal Univ Nat Sci, 2007, 23(6): 19-23 · Zbl 1145.16306 |
| [7] | Christopher C G. Tensor products of Young modules. J Algebra, 2012, 366: 12-34 · Zbl 1280.20012 · doi:10.1016/j.jalgebra.2012.05.014 |
| [8] | Eilenberg S, Rosenberg A, Zalinsky D. On the dimension of modules and algebras VIII. Nagoya Math J, 1957, 12: 71-93 · Zbl 0212.05901 · doi:10.1017/S0027763000021954 |
| [9] | Happel D. Triangulated Categories in the Representation Theory of Finite Dimensional Algebras. London Math Soc Lecture Note Ser, 119. Cambridge: Cambridge Univ Press, 1988 · Zbl 0635.16017 · doi:10.1017/CBO9780511629228 |
| [10] | Happel D, Ringel C. Tilted algebras. Trans Amer Math Soc, 1982, 274: 399-443 · Zbl 0503.16024 · doi:10.1090/S0002-9947-1982-0675063-2 |
| [11] | Hugel A L, Coelho F U. In finitely generated tilting modules of finite projective dimension. Forum Math, 2001, 13: 239-250 · Zbl 0984.16009 · doi:10.1515/form.2001.006 |
| [12] | Miyashita Y. Tilting modules of finite projective dimension. Math Z, 1986, 193: 113-146 · Zbl 0578.16015 · doi:10.1007/BF01163359 |
| [13] | Rickard J. Morita theory for derived categories. J Lond Math Soc, 1989, 39: 436-456 · Zbl 0642.16034 · doi:10.1112/jlms/s2-39.3.436 |
| [14] | Rotman J J. An Introduction to Homological Algebra. 2nd ed. New York: Springer, 2009 · Zbl 1157.18001 · doi:10.1007/b98977 |
| [15] | Yang J W. Tensor products of strongly graded vertex algebras and their modules. J Pure Appl Algebra, 2013, 217(2): 348-363 · Zbl 1281.17032 · doi:10.1016/j.jpaa.2012.06.021 |
| [16] | Zhou B X. On the tensor product of left modules and their homological dimensions. J Math Res Exposition, 1981, 1: 17-24 · Zbl 0581.13012 |
| [17] | Zhou B X. Homological Algebra. Beijing: Science Press, 1988 (in Chinese) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
