τ-Tilting Modules Over One-Point Extensions by a Projective Module
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Abstract
Let A be the one point extension of an algebra B by a projective B-module. We prove that the extension of a given support τ-tilting B-module is a support τ-tilting A-module; and, conversely, the restriction of a given support τ-tilting A-module is a support τ-tilting B-module. Moreover, we prove that there exists a full embedding of quivers between the corresponding poset of support τ-tilting modules.
Keywords
One-point extension Tilting modules Poset τ-tilting modulesMathematics Subject Classification (2010)
16G20 16E10 16E30Preview
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Notes
Acknowledgments
The author thankfully acknowledge partial support from CONICET and from Universidad Nacional de Mar del Plata, Argentina. The results of this article are part of the PhD thesis of the author under the supervision of Sonia Trepode and Claudia Chaio. She is grateful to them for their constant support and helpful discussions.
References
- 1.Adachi, T., Iyama, O., Reiten, I.: τ-tilting theory. Compos. Math. 150(03), 415–452 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
- 2.Assem, I., Happel, D., Trepode, S.: Extending tilting modules to one-point extensions by projectives. Comm. Algebra 35(10), 2983–3006 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
- 3.Assem, I., Simson, D., Skowronski, A.: Elements of the Representation Theory Of Associative Algebras. London Math. Soc. Student Texts 65. Cambridge University Press, Cambridge (2006)CrossRefGoogle Scholar
- 4.Auslander, M., Smalø, S.O.: Almost split sequences in subcategories. J. Algebra 69(2), 426–454 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
- 5.Bongartz, K.: Tilted Algebras. Proc. ICRA III (Puebla 1980), Lecture Notes in Math. No. 903, Springer-Verlar, 2936Google Scholar
- 6.Brenner, S., Butler, M.C.R.: Generalizations of Bernstein-Gelfand-Ponomarev Reflection Functors. Lecture Notes in Math, vol. 839, pp 103–169. Springer, Berlin (1980)Google Scholar
- 7.Happel, D., Ringel, C.M.: Tilted algebras. Trans. Amer. Math. Soc. 274, 399–443 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
- 8.Jasso, G.: Reduction of τ-tilting modules and torsion pairs. International Mathematics Research Notices, rnu163 (2014)Google Scholar
- 9.Psaroudakis, C.: Homological theory of recollements of abelian categories. J. Algebra 398, 63–110 (2014)MathSciNetCrossRefzbMATHGoogle Scholar