Canonical bimodules and dominant dimension

Fang, Ming; Kerner, Otto; Yamagata, Kunio

doi:10.1090/tran/6976

Canonical bimodules and dominant dimension
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by Ming Fang, Otto Kerner and Kunio Yamagata PDF

Trans. Amer. Math. Soc. 370 (2018), 847-872 Request permission

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

C. Amiot, Preprojective algebras and Calabi-Yau duality, arXiv:1404.4764v1.
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, DOI 10.1017/CBO9780511623608
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
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, DOI 10.1090/conm/331/05901
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, DOI 10.1080/00927878708823425
Henri Cartan and Samuel Eilenberg, Homological algebra, Princeton University Press, Princeton, N. J., 1956. MR 0077480
R. R. Colby and K. R. Fuller, Exactness of the double dual, Proc. Amer. Math. Soc. 82 (1981), no. 4, 521–526. MR 614871, DOI 10.1090/S0002-9939-1981-0614871-5
R. R. Colby, Nakayama’s conjecture and the double dual functors, J. Algebra 94 (1985), no. 2, 546–557. MR 792969, DOI 10.1016/0021-8693(85)90198-X
Ming Fang, Permanents, Doty coalgebras and dominant dimension of Schur algebras, Adv. Math. 264 (2014), 155–182. MR 3250282, DOI 10.1016/j.aim.2014.07.005
Ming Fang and Steffen Koenig, Schur functors and dominant dimension, Trans. Amer. Math. Soc. 363 (2011), no. 3, 1555–1576. MR 2737277, DOI 10.1090/S0002-9947-2010-05177-3
Ming Fang and Steffen Koenig, Endomorphism algebras of generators over symmetric algebras, J. Algebra 332 (2011), 428–433. MR 2774695, DOI 10.1016/j.jalgebra.2011.02.031
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, DOI 10.1090/tran/6504
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, DOI 10.1006/jabr.2000.8726
Otto Kerner and Kunio Yamagata, Morita algebras, J. Algebra 382 (2013), 185–202. MR 3034479, DOI 10.1016/j.jalgebra.2013.02.013
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 96700
Kiiti Morita, Duality in $\textrm {QF}-3$ rings, Math. Z. 108 (1969), 237–252. MR 241470, DOI 10.1007/BF01112025
Bruno J. Müller, The classification of algebras by dominant dimension, Canadian J. Math. 20 (1968), 398–409. MR 224656, DOI 10.4153/CJM-1968-037-9
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
Joseph J. Rotman, An introduction to homological algebra, 2nd ed., Universitext, Springer, New York, 2009. MR 2455920, DOI 10.1007/b98977
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, DOI 10.17323/1609-4514-2008-8-1-119-158
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, DOI 10.1090/S0002-9947-99-02362-4
Andrzej Skowroński and Kunio Yamagata, Frobenius algebras. I, EMS Textbooks in Mathematics, European Mathematical Society (EMS), Zürich, 2011. Basic representation theory. MR 2894798, DOI 10.4171/102
Hiroyuki Tachikawa, On dominant dimensions of $\textrm {QF}$-3 algebras, Trans. Amer. Math. Soc. 112 (1964), 249–266. MR 161888, DOI 10.1090/S0002-9947-1964-0161888-8
Hiroyuki Tachikawa, Double centralizers and dominant dimensions, Math. Z. 116 (1970), 79–88. MR 265407, DOI 10.1007/BF01110189
Hiroyuki Tachikawa, Quasi-Frobenius rings and generalizations. $\textrm {QF}-3$ and $\textrm {QF}-1$ rings, Lecture Notes in Mathematics, Vol. 351, Springer-Verlag, Berlin-New York, 1973. Notes by Claus Michael Ringel. MR 0349740
Kunio Yamagata, Frobenius algebras, Handbook of algebra, Vol. 1, Handb. Algebr., vol. 1, Elsevier/North-Holland, Amsterdam, 1996, pp. 841–887. MR 1421820, DOI 10.1016/S1570-7954(96)80028-3
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, DOI 10.1090/conm/607/12094