Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Canonical bimodules and dominant dimension


Authors: Ming Fang, Otto Kerner and Kunio Yamagata
Journal: Trans. Amer. Math. Soc. 370 (2018), 847-872
MSC (2010): Primary 16D20, 16D40, 16D50, 16E10
DOI: https://doi.org/10.1090/tran/6976
Published electronically: July 13, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: For a finite dimensional algebra $ A$ over a field $ k$, the inherent $ A$-bimodules which include $ A$ and its $ k$-dual $ \mathrm {D}(A)$, as well as those derived from them by iteratively taking their left or right $ A$-duals or higher extensions, are crucial in many considerations. We study the properties of these bimodules, mainly of $ \mathrm {Hom}_A(\mathrm {D}(A),A)$ (called the canonical $ A$-bimodule), and utilize them to provide new characterizations of Morita algebras and the dominant dimension of $ A$.


References [Enhancements On Off] (What's this?)

  • [1] C. Amiot, Preprojective algebras and Calabi-Yau duality, arXiv:1404.4764v1.
  • [2] Maurice Auslander, Idun Reiten, and Sverre O. Smalø, Representation theory of Artin algebras, Cambridge Studies in Advanced Mathematics, vol. 36, Cambridge University Press, Cambridge, 1995. MR 1314422
  • [3] Maurice Auslander and Mark Bridger, Stable module theory, Memoirs of the American Mathematical Society, No. 94, American Mathematical Society, Providence, R.I., 1969. MR 0269685
  • [4] Ragnar-Olaf Buchweitz, Morita contexts, idempotents, and Hochschild cohomology--with applications to invariant rings, Commutative algebra (Grenoble/Lyon, 2001) Contemp. Math., vol. 331, Amer. Math. Soc., Providence, RI, 2003, pp. 25-53. MR 2011764, https://doi.org/10.1090/conm/331/05901
  • [5] Dagmar Baer, Werner Geigle, and Helmut Lenzing, The preprojective algebra of a tame hereditary Artin algebra, Comm. Algebra 15 (1987), no. 1-2, 425-457. MR 876985, https://doi.org/10.1080/00927878708823425
  • [6] Henri Cartan and Samuel Eilenberg, Homological algebra, Princeton University Press, Princeton, N. J., 1956. MR 0077480
  • [7] R. R. Colby and K. R. Fuller, Exactness of the double dual, Proc. Amer. Math. Soc. 82 (1981), no. 4, 521-526. MR 614871, https://doi.org/10.2307/2043764
  • [8] R. R. Colby, Nakayama's conjecture and the double dual functors, J. Algebra 94 (1985), no. 2, 546-557. MR 792969, https://doi.org/10.1016/0021-8693(85)90198-X
  • [9] Ming Fang, Permanents, Doty coalgebras and dominant dimension of Schur algebras, Adv. Math. 264 (2014), 155-182. MR 3250282, https://doi.org/10.1016/j.aim.2014.07.005
  • [10] Ming Fang and Steffen Koenig, Schur functors and dominant dimension, Trans. Amer. Math. Soc. 363 (2011), no. 3, 1555-1576. MR 2737277, https://doi.org/10.1090/S0002-9947-2010-05177-3
  • [11] Ming Fang and Steffen Koenig, Endomorphism algebras of generators over symmetric algebras, J. Algebra 332 (2011), 428-433. MR 2774695, https://doi.org/10.1016/j.jalgebra.2011.02.031
  • [12] Ming Fang and Steffen Koenig, Gendo-symmetric algebras, canonical comultiplication, bar cocomplex and dominant dimension, Trans. Amer. Math. Soc. 368 (2016), no. 7, 5037-5055. MR 3456170, https://doi.org/10.1090/tran/6504
  • [13] Steffen König, Inger Heidi Slungård, and Changchang Xi, Double centralizer properties, dominant dimension, and tilting modules, J. Algebra 240 (2001), no. 1, 393-412. MR 1830559, https://doi.org/10.1006/jabr.2000.8726
  • [14] Otto Kerner and Kunio Yamagata, Morita algebras, J. Algebra 382 (2013), 185-202. MR 3034479, https://doi.org/10.1016/j.jalgebra.2013.02.013
  • [15] Kiiti Morita, Duality for modules and its applications to the theory of rings with minimum condition, Sci. Rep. Tokyo Kyoiku Daigaku Sect. A 6 (1958), 83-142. MR 0096700
  • [16] Kiiti Morita, Duality in $ {\rm QF}-3$ rings, Math. Z. 108 (1969), 237-252. MR 0241470, https://doi.org/10.1007/BF01112025
  • [17] Bruno J. Müller, The classification of algebras by dominant dimension, Canad. J. Math. 20 (1968), 398-409. MR 0224656, https://doi.org/10.4153/CJM-1968-037-9
  • [18] Claus Michael Ringel, The preprojective algebra of a quiver, Algebras and modules, II (Geiranger, 1996) CMS Conf. Proc., vol. 24, Amer. Math. Soc., Providence, RI, 1998, pp. 467-480. MR 1648647
  • [19] Joseph J. Rotman, An introduction to homological algebra, 2nd ed., Universitext, Springer, New York, 2009. MR 2455920
  • [20] Raphaël Rouquier, $ q$-Schur algebras and complex reflection groups, Mosc. Math. J. 8 (2008), no. 1, 119-158, 184 (English, with English and Russian summaries). MR 2422270
  • [21] Andrzej Skowroński and Kunio Yamagata, Galois coverings of selfinjective algebras by repetitive algebras, Trans. Amer. Math. Soc. 351 (1999), no. 2, 715-734. MR 1615962, https://doi.org/10.1090/S0002-9947-99-02362-4
  • [22] Andrzej Skowroński and Kunio Yamagata, Frobenius algebras. I: Basic representation theory, EMS Textbooks in Mathematics, European Mathematical Society (EMS), Zürich, 2011. MR 2894798
  • [23] Hiroyuki Tachikawa, On dominant dimensions of $ {\rm QF}$-3 algebras, Trans. Amer. Math. Soc. 112 (1964), 249-266. MR 0161888, https://doi.org/10.2307/1994293
  • [24] Hiroyuki Tachikawa, Double centralizers and dominant dimensions, Math. Z. 116 (1970), 79-88. MR 0265407, https://doi.org/10.1007/BF01110189
  • [25] Hiroyuki Tachikawa, Quasi-Frobenius rings and generalizations. $ {\rm QF}-3$ and $ {\rm QF}-1$ rings, Lecture Notes in Mathematics, Vol. 351, Springer-Verlag, Berlin-New York, 1973. MR 0349740
  • [26] Kunio Yamagata, Frobenius algebras, Handbook of algebra, Vol. 1, Handb. Algebr., vol. 1, Elsevier/North-Holland, Amsterdam, 1996, pp. 841-887. MR 1421820, https://doi.org/10.1016/S1570-7954(96)80028-3
  • [27] Kunio Yamagata and Otto Kerner, Morita theory, revisited, Expository lectures on representation theory, Contemp. Math., vol. 607, Amer. Math. Soc., Providence, RI, 2014, pp. 85-96. MR 3204867, https://doi.org/10.1090/conm/607/12094

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 16D20, 16D40, 16D50, 16E10

Retrieve articles in all journals with MSC (2010): 16D20, 16D40, 16D50, 16E10


Additional Information

Ming Fang
Affiliation: Academy of Mathematics and Systems Science, Chinese Academy of Sciences, 100190 – and – School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, People’s Republic of China
Email: fming@amss.ac.cn

Otto Kerner
Affiliation: Mathematisches Institut, Heinrich-Heine-Universität, 40225, Düsseldorf, Germany
Email: kerner@math.uni-duesseldorf.de

Kunio Yamagata
Affiliation: Institute of Engineering, Tokyo University of Agriculture and Technology, Tokyo 184-8588, Japan
Email: yamagata@cc.tuat.ac.jp

DOI: https://doi.org/10.1090/tran/6976
Keywords: Bimodule, dominant dimension, Morita algebra
Received by editor(s): May 28, 2015
Received by editor(s) in revised form: April 30, 2016
Published electronically: July 13, 2017
Additional Notes: The first-named author’s research was supported by Natural Science Foundation of China (No. 11271318 and No. 11471315). The third-named author’s research was supported by JSPS KAKENHI (No. 25400036 and No. 16K05091)
Dedicated: Dedicated to C. M. Ringel on the occasion of his 70th birthday
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society