Algebras and Representation Theory

, Volume 21, Issue 4, pp 769–786

# τ-Tilting Modules Over One-Point Extensions by a Projective Module

• Pamela Suarez
Article

## 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 modules

## Mathematics Subject Classification (2010)

16G20 16E10 16E30

## 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. 1.
Adachi, T., Iyama, O., Reiten, I.: τ-tilting theory. Compos. Math. 150(03), 415–452 (2014)
2. 2.
Assem, I., Happel, D., Trepode, S.: Extending tilting modules to one-point extensions by projectives. Comm. Algebra 35(10), 2983–3006 (2007)
3. 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)
4. 4.
Auslander, M., Smalø, S.O.: Almost split sequences in subcategories. J. Algebra 69(2), 426–454 (1981)
5. 5.
Bongartz, K.: Tilted Algebras. Proc. ICRA III (Puebla 1980), Lecture Notes in Math. No. 903, Springer-Verlar, 2936Google Scholar
6. 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. 7.
Happel, D., Ringel, C.M.: Tilted algebras. Trans. Amer. Math. Soc. 274, 399–443 (1982)
8. 8.
Jasso, G.: Reduction of τ-tilting modules and torsion pairs. International Mathematics Research Notices, rnu163 (2014)Google Scholar
9. 9.
Psaroudakis, C.: Homological theory of recollements of abelian categories. J. Algebra 398, 63–110 (2014)