Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 12 (2016), 092, 24 pages      arXiv:1504.03548      https://doi.org/10.3842/SIGMA.2016.092

Further Properties and Applications of Koszul Pairs

Adrian Manea and Dragoş Ştefan
Faculty of Mathematics and Computer Science, University of Bucharest, 14 Academiei Str., Bucharest Ro-010014, Romania

Received May 19, 2016, in final form September 08, 2016; Published online September 14, 2016

Abstract
Koszul pairs were introduced in [arXiv:1011.4243] as an instrument for the study of Koszul rings. In this paper, we continue the enquiry of such pairs, focusing on the description of the second component, as a follow-up of the study in [arXiv:1605.05458]. As such, we introduce Koszul corings and prove several equivalent characterizations for them. As applications, in the case of locally finite $R$-rings, we show that a graded $R$-ring is Koszul if and only if its left (or right) graded dual coring is Koszul. Finally, for finite graded posets, we obtain that the respective incidence ring is Koszul if and only if the incidence coring is so.

Key words: Koszul rings; Koszul corings; Koszul pairs; incidence (co)ring of a poset.

pdf (471 kb)   tex (32 kb)

References

  1. Beilinson A., Ginzburg V., Soergel W., Koszul duality patterns in representation theory, J. Amer. Math. Soc. 9 (1996), 473-527.
  2. Berger R., Koszulity for nonquadratic algebras, J. Algebra 239 (2001), 705-734.
  3. Brzezinski T., Wisbauer R., Corings and comodules, London Mathematical Society Lecture Note Series, Vol. 309, Cambridge University Press, Cambridge, 2003.
  4. Green E.L., Reiten I., Solberg Ø., Dualities on generalized Koszul algebras, Mem. Amer. Math. Soc. 159 (2002), xvi+67 pages.
  5. Jara Martínez P., López Peña J., Ştefan D., Koszul pairs and applications, J. Noncommut. Geom., to appear, arXiv:1011.4243.
  6. Keller B., Koszul duality and coderived categories (after K. Lefèvre), available at https://webusers.imj-prg.fr/~bernhard.keller/publ/kdcabs.html.
  7. Madsen D.O., On a common generalization of Koszul duality and tilting equivalence, Adv. Math. 227 (2011), 2327-2348, arXiv:1007.3282.
  8. Madsen D.O., Quasi-hereditary algebras and generalized Koszul duality, J. Algebra 395 (2013), 96-110, arXiv:1201.0441.
  9. Manea A., Ştefan D., On Koszulity of finite graded posets, J. Algebra Appl., to appear, arXiv:1605.05458.
  10. May J.P., Bialgebras and Hopf algebras, available at http://www.math.uchicago.edu/~may/TQFT/HopfAll.pdf.
  11. Mazorchuk V., Ovsienko S., Stroppel C., Quadratic duals, Koszul dual functors, and applications, Trans. Amer. Math. Soc. 361 (2009), 1129-1172, math.RT/0603475.
  12. Piontkovski D., Graded algebras and their differentially graded extensions, J. Math. Sci. 142 (2007), 2267-2301.
  13. Polishchuk A., Positselski L., Quadratic algebras, University Lecture Series, Vol. 37, Amer. Math. Soc., Providence, RI, 2005.
  14. Polo P., On Cohen-Macaulay posets, Koszul algebras and certain modules associated to Schubert varieties, Bull. London Math. Soc. 27 (1995), 425-434.
  15. Priddy S.B., Koszul resolutions, Trans. Amer. Math. Soc. 152 (1970), 39-60.
  16. Reiner V., Stamate D.I., Koszul incidence algebras, affine semigroups, and Stanley-Reisner ideals, Adv. Math. 224 (2010), 2312-2345, arXiv:0904.1683.
  17. Woodcock D., Cohen-Macaulay complexes and Koszul rings, J. London Math. Soc. 57 (1998), 398-410.

Previous article  Next article   Contents of Volume 12 (2016)