Abstract
A pair (A, P) is called a cover of EndA(P)op if the Schur functor HomA(P,−) is fully faithful on the full subcategory of projective A-modules, for a given projective A-module P. By definition, Morita algebras are the covers of self-injective algebras and then P is a faithful projective-injective module. Conversely, we show that A is a Morita algebra and EndA(P)op is self-injective whenever (A, P) is a cover of EndA(P)op for a faithful projective-injective module P.
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References
Chan, A., Marczinzik, R.: On representation-finite gendo-symmetric biserial algebras. Algebr. Represent. Theory 22(1), 141–176 (2019)
Cruz, T.: Schur-Weyl duality over commutative rings. Commun. Algebra 47(4), 1619–1628 (2019)
Dao, H., Iyama, O., Takahashi, R., Vial, C.: Non-commutative resolutions and Grothendieck groups. J. Noncommut. Geom. 9(1), 21–34 (2015)
Fang, M., Hu, W., Koenig, S.: On derived equivalences and homological dimensions. J. Reine Angew. Math. 770, 59–85 (2021)
Fang, M., Kerner, O., Yamagata, K.: Canonical bimodules and dominant dimension. Trans. Am. Math. Soc. 370(2), 847–872 (2018)
Green, J.A.: Polynomial Representations of GLn, volume 830 of Lecture Notes in Mathematics. Springer, Berlin (1980)
Koenig, S., Slungård, I.H., Xi, C.: Double centralizer properties, dominant dimension, and tilting modules. J. Algebra 240(1), 393–412 (2001)
Kerner, O., Yamagata, K.: Morita algebras. J. Algebra 382, 185–202 (2013)
Morita, K.: Duality for modules and its applications to the theory of rings with minimum condition. Sci. Rep. Tokyo Kyoiku Daigaku, Sect. A 6, 83–142 (1958)
Mueller, B.J.: The classification of algebras by dominant dimension. Can. J. Math. 20, 398–409 (1968)
Rouquier, R.: q-Schur algebras and complex reflection groups. Mosc. Math. J. 8(1), 119–158 (2008)
Soergel, W.: Kategorie \(\mathcal {O}\), perverse Garben und Moduln über den Koinvarianten zur Weylgruppe. J. Am. Math. Soc. 3(2), 421–445 (1990)
Tachikawa, H.: Quasi-Frobenius rings and generalizations. QF-3 and QF-1 rings. Notes by Claus Michael Ringel, vol. 351. Springer, Cham (1973)
Thrall, R.M.: Some generalizations of quasi-Frobenius algebras. Trans. Am. Math. Soc. 64, 173–183 (1948)
Yamagata, K., Kerner, O.: Morita theory, revisited. In: Expository lectures on representation theory. Maurice Auslander distinguished lectures and international conference, Woods Hole Oceanographic Institute, Quisset Campus, Falmouth, MA, USA, April 25–30, 2012, pp 85–96. American Mathematical Society (AMS), Providence (2014)
Acknowledgments
The results of this paper are contained in the author’s forthcoming PhD thesis, financially supported by Studienstiftung des Deutschen Volkes. The author would like to thank Steffen Koenig for all his comments and suggestions towards improving this manuscript.
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Presented by: Christof Geiss
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Cruz, T. A Characterisation of Morita Algebras in Terms of Covers. Algebr Represent Theor 25, 1197–1206 (2022). https://doi.org/10.1007/s10468-021-10064-8
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DOI: https://doi.org/10.1007/s10468-021-10064-8